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^ WIpEjyER 






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Logica est ars artinm et ecientia scientiarum. — Sootvs. 




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Introduction 1 

I. OfTbbms 4 

II. Of FBOFosrnoNS 8 


IV. Of Combination of Tbbhs 14 

V. Of Skpabation of Tbbhs 22 

VI. Of Plural Terms 25 

Vn. Of Negative Propositions 29 

VIII. Of Contbabt Tebms 31 

IX. Of Contbaby Altebnatiyes . ... .36 

X. Of Contbaby Tebms in Propositions . .39 

XI. Of Indirect Infbbbnce 42 

XII. Of Belation to Common Logic 53 

XIII. Examples of the Method 58 


XV. Ebmabks on Boole's System, and on the Belation of 

Logic and Mathematics ... 75 

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It is the purpose of this work to show that Logic 
assumes a new degree of simplicity, precision, 
generality, and power, when comparison in quality 
is treated apart from any reference to quantity. 

1. It is familiarly known to logicians that a Extent and 

term must be considered with respect both to the «w^««^ of 

individual things it denotes, and the qualities, cir- 
cumstances, or attributes it connoteSy or implies as 
belonging to those things. The number of indi- 
viduals denoted forms the breadth or extent of the 
meaning of the term ; the qualities or attributes 
connoted form the depth, comprehension, or intent, 
of the meaning of the term. The extent and in- 
tent of meaning, however, are closely related, and 
in a reciprocal manner. The more numerous the 
qualities connoted by a term, the fewer in general 
the individuals which it can denote ; the one di- 
mension, so to speak, of the meaning being given, 
the other follows, and cannot be given or taken 
at will. 

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2. Logicians have generally thought that a 
proposition must express the relations of extent 
and intent of the terms at one and the same 
time, and as regarded in the same light. The 
systems of logic deduced from such a view, when 
compared .with the system which may otherwise 
be had, seem to lack simplicity and generality. 

3. It is here held that a proposition expresses 
the result of a comparison and judgment of the 
sameness or difference of meaning of terms, either 
in intent or extent of meaning* The judgment in 
the one dimension of meaning, however, is not 
independent of the judgment in the other dimen- 
sion. It is only then judgment and reasoning in 
one dimension which is properly expressed in a 
simple system. Judgment and reasoning in the 
other dimension will be and must be implied. It 
may be expressed in a numerical or quantitative 
system corresponding to the qualitative system, 
but its expression in the same system destroys! 

I do not wish to express any opinion here as to 
the nature of a system of logic in extent, nor as 
to its precise connection vdth the pure system of 
logic of quality. 

4. Keasoning in quality and quantity, in intent 
or extent of meaning, being considered apart, it 
seems obvious that the comparison of things in 
quality, with respect to all their points of same- 
ness and diflference, gives the primary and most 
general system of reasoning. It even seems likely 
that such a system must comprehend all possible 
and conceivable kinds of reasoning, since it treats 

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of any and every way in which things may be 
same or different. All reasoning is probably 
founded on the laws of sameness and difference 
which form the basis of the following system. 

6. My present task, however, is to show that Present 
all and more than all the ordinary processes of 
logic may he combined in a system founded on 
comparison of quality only^ without reference to 
logical quantity. 

6. Before proceeding I have to acknowledge Belation to 
that in a considerable degree this system is foimded -j. * 
on that of Prof. Boole, as stated in his admirable 
and highly original Mathematical Analysis of 
Logic* The forms of my system may, in fact, 
be reached by divesting his system of a mathe- 
matical dress, which, to say the least, is not 
essential to it. The system being restored to its 
proper simplicity, it may be inferred, not that 
Logic is a part of Mathematics, as is almost im- 
plied in Prof. Boole's writings, but that the Ma- 
thematics are rather derivatives of Logic. All 
the interesting analogies or samenesses of logical 
and mathematical reasoning which may be pointed 
out, are surely reversed by making Logic the 
dependent of Mathematics. 

* Investigation of the Laws of Thought. By George 
Boole, LL.D. London, 1854. Frequent reference will be 
made to this work in the following pages. 

B 2 

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Of things 
and their 

of name. 


7. Pure logic arises from a comparison of things 
as to their sameness or difference in any guality or 
circumstance whatever. , i 

In discourse we refer to things by the aid 
of marks, names, or terms, which are also, as it 
were, the handles by which the mind grasps and 
retains its thoughts about things. Thus correct 
thought about things becomes in ^scourse the 
correct use of names. Logic, while treating only 
of names, ascertaining the relations of sameness 
and difference of their meanings, treats indirectly, ^ , 
as alone it can, of the samenesses and differences^V^> 
of thmgs. \ 

8. A term taken in intent has for its meaning '. 
the whole infinite series of qualities and circum^ ^' 
stances which a thing possesses. Of these qualities 
or circimistances some may be known and form 
the description or definition of the meaning ; the 
infinite remainder are vmknown. 

Among the circumstances, indeed, of a thing, is 
the fact of its being denoted by a given name, but 
we may speak of a thing, of which only the name 
is known, as having a name of unknoivn meaning. 

The meaning of every name, then, is either un- 

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known or more or leas known. But we may speak 
of a term that is more or leas known as being 
simply kjwwn. 

9. Among the qualities and circumstances of a Qualities 
thing is to be counted everything that may be said ^^Jj^/** 
of it, affirmatively or negatively. Any possible 
quality or circumstance that can be thought of 

either does or does not apply to any given thing, 
and therefore forms, either affirmatively or nega- 
tively, a quality or circumstance of the thing. 
Concerning anything, then, there may be an in- 
finit|@ number of statements made, or qualities 

10. When we assign a name to a thing, with Bdation of 
knowledge of, and regard to, certain of its qualities ^''^^*'^*' 
or circumstances, that name is equally the name 

of anything el^e of exactly the same known quali- 
ties and circumstances. For there is nothing in the 
name to determine it to the one thing rather than 
the other. Any name, then, must be the name in 
extent of anything, and of all things agreeing in 
the qualities or circumstances which form its 
known meaning in intent, and in this system. 

11. Though it is well to point out that all our Present 
names or terms bear a universal quantity when. ^"^^^' 
regarded in extent, it must be understood, and 
constantly borne in mind, that ftirther reference 

to the meaning of a term in extent or quantity of 
individuals, is excluded in these pages. 

jThe primary and only present meaning of a 
name or term is a certain set of qualities, attributes^ * 
properties, or circumstances^ of a thing unknown 
or partly known. 

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of our 



of same- 
ness of 

12. Term will be used to mean name, or any 
combination of names and words describing the 
qualities and circumstances of a thing. 

13. The terms of this system may be made to 
express any combination of samenesses and diffe- 
rences in quality, kind, attribute, circumstance, 
number, magnitude, degree, quantity, opposition, 
or distance in time or space. A term may thus 
represent the qualities of a thing or person in all 
the complexity of real existence, so well and fully 
defined that we cannot suppose there are, or 
are likely to be, two things the same in so many 
circumstances. Such a term would correspond to 
the singular, proper, non-attrihutive, or non-con- 
notative names of the old logic. Such names are 
accordingly by no means excluded from this sys- 
tem ; and it is here held that the old distinction 
oi connotative and nxm-connotative names is wholly 
erroneous and unfounded. If there is any dis- 
tinction to be drawn, it is that singular, proper, or 
so-called non-connotative terms, are more full of 
connotation or meaning in intent or quality than 
others, instead of being devoid of such meaning. 

14. As logic only considers the relations of 
meaning of terms, as expressed within a piece of 
reasoning, the special meaning of any term is of 
no account, provided that the same term have 
the same meaning throughout any one piece of 

Thus, instead of the nouns and adjectives, to 
each of which a special meaning is assigned in 
common discourse, we shall use certain letters, 
A, B, C, D, . . . • U, V . . . each standing 

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for a special term, or a definite meaning, and 
for any term or meaning^ always under the above 

16. Our terms, A, B, C, like the Term 

terms of common discourse, may be either known' u^f^jcnown, 
or imknown in meaning. It is the work of logic 
to show what relations of sameness and difference " 
between unknown and known terms may make the 
unknown terms known. 

Were it not to explain ignotum per ignotiv^s^ we 
might say that logic is the algebra of kind or 
quality^ the calculus of known and unknown quali^ 
tieSy as algebra (more strictly speaking universal 
arithmetic, which does not recognise essentially 
negative quantities) is the calculus of known and 
imknown quantities. 

16. Let it be borne in mind that the letters A, B, Symbols of 
C, &c., as well as the marks +, 0, and =, after- ^fieaning, 
wards to be introduced, are in no way mysterious 
symbols. The term A, for instaace, is merely a 
convenient abbreviation for any ordinary term of 
language, or set of terms, such as Bed, or the Lords 
Commissioners for executing the office of the Lord 
High Admiral of England, 

Again, + is merely a mark substituted for the 
sake of clearness, for the conjunctions and, either, 
or, &c., of common language. The mark = is 
merely the copula is, or is same as, or some 
equivalent. The meaning of 0, whatever it ex- 
actly be, may also be expressed in words. There 
is consequently nothing more symbolic or myste- 
rious in this system than in conamon language. 

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tion dc' 



Truth and 

of affirma" 
Uve propo' 

17. A proposition is a statement of the samenesst 
or difference of meaning of two terms, that is, of the 
sameness or difference of the qualities and circum- 
stances connoted by each term. 

18. According as a proposition states sameness 
or difference^ it is called affirmative or negative, 

19. It is the purpose or use of a proposition to 
make known the meaning of a term that is otherwise 

20. A proposition is said to be true when the 
meanings of its terms are same or different, as 
stated; otherwise it is called false or untrue. As 
logic deals with things only through terms, it 
cannot ascertain whether a proposition is true or 
false, but only whether two or more propositions 
are or are not true together, imder the condition 
of meaning of terms (§ 14). 

21. We denote by the copula is, or by the mark 
=, the sameness of meaning of the terms on the 
two sides of a proposition. 

For the present we shall speak only of affirma- 
tive propositions, which are of superior importance ; 
and when not otherwise specified, proposition maj 
be taken to mean affirmative proposition. 

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22. A proposition is simply convertible. The Conversion 
propositions A=B and B=A axe the same ^fP^^^^P*^' 
statement; either of the terms A and B is the 

same in meaning as the other, nndistinguiBhable 
except in name. 

This simple conversion comprehends both the 
simple conversion, and conversio per accidens of the 
school logic. 

23. One proposition and one known term may One term 

make known one unknown term. known 

_ . -w^ « , ■« 1 . from one 

From A=B, so far as we know B, that is, proposi- 

know its meaning, we can learn A ; so ^ as we ^*^^- 
know A, we can learn B. 

We thus know samely of both sides of a propo- 
sition whatever we know of either. The same 
might be said of uncertain or obscure knowledge. 

24. A proposition between any two terms of Useless 
which the meanings are otherwise known as same ^^ *jf^^f" 
or different, is useless. For it cannot serve the sitions 
purpose of a proposition (§ 19). Such is any "^^w^^^^- 
proposition between a term and itself, as A=A, 

B=:B (§ 14). These useless propositions are 
called Identical, They state the condition of all 
reasoning, but we know it without the statement. 
A proposition repeated, or a converted propo- 
sition (§ 22), is also useless, except for the mere 
convenience of memory, or ready apprehension. 

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Law of 25. It is in the nature of thought and things, 

sameness. ^^^ things which are same as the same thing are 
the same as each other. 
More briefly — Same as same are same. 
Hence the first law of logic — that terms which 
are same in,meaning as the same term^ are the same 
in meaning as each other. 

This law, it is obvious, is analogous to Euclid's 
first aadom, or common notion, that things which 
are equal to the same thing, are equal to each other. 
Things are called equal which are same in magni- 
tude, but what is true of such sameness, is also 
true of sameness in any way in which things may 
be same or different. Euclid's geometrical law is 
but one case of the general law. 
Meaning of 26. Logic proceeds by laws, and is bound by 
kiws of them. For logic must treat names as thought 
treats things. And the laws of logic state certain 
samenesses or uniformities in our ways of thinking, 
and are of self-evident truth. 
Direct in- 27. When two affirmative propositions are same 
shown and *^* ^^^ member of each, the other members may be 
defined. stated to be same. 

From A=B, B=C, which are the same 

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in the member B, we may form the new propo- 
sition A=C. For A and C being each stated to 
be the same as B, may by the law of sameness be 
stated to be the same as each other. 

A proposition got by the Law of Sameness is 
said to be got by direct inference^ and is called a 
direct inferentj or, in common language, a direct 

28. Propositions from wnich an inference is Premise 
drawn are called premises^ and are given or taken '^^^ ' 
as the basis of reasoning. Logic is not concerned 

with the truth or falsity of premises or inference, 
except as regards the truth or falsity of one with 
the other. (§§ 20, 37.) 

29. An expression for a term consists of any Expression 
other term which by premises we know to be the "^^ ' 
same in meaning with that term. 

30. In inferring a new proposition from two Elimina- 
premises we are said to eliminate or remove j^^^^, 
the member which is the same in the two pre- 

From two premises we may eliminate only one 
term, and infer one new proposition. By saying 
that we may, it is not meant that we always 

31. Propositions are said to be related to each Related 
other which have a same or common member, or fjj^ 
which are so related to other propositions so re- d^md. 
lated ; and so on. 

In other words, any two propositions are related 
which form part of a series or chain of propositions, 
in which each proposition is related to the adjoin- 
ing ones or one. 

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terms de* 

Use of 

Series oj 

NuTnher of 
terms and 

terms and 

32. Terms are said to be related which occirf 
in one same, or in any related propositions. 

33. From two related premises and one known 
term we may learn two unknown terms, and not 

From A=B and B=C, we learn any two of 
A, B, C, when the third is known. 

34. From any series of related premises, and 
one known term we may learn as many unknoivn 
terms as there are premises. Thus, from A = B 
= C=D=E=F, we may learn any five terms 
when the sixth is known. For each useful propo- 
sition may render one unknown term known 
(§ 19). Between each two adjoining premises 
one term may be eliminated, becoming known in 
one premise, and rendering another term known 
in the other. There must at last remain a single 
proposition containing two terms, each of which 
occurs only in one premise. 

35. The number of related premises must be 
one less than the number of different terms. If it 
be still less, the propositions cannot be all related ; 
if it be greater, some of the premises must be 
useless, because they must lie between terms 
otherwise known to be same by inference. 

It will be remarked that systems of mathe- 
matical propositions or equations with known and 
imknown qualities are perfectly analogous in their 
properties to logical propositions. 

36. When a related premise contains a term 
or member not relevant to the purpose of the 
reasoning, this term is eliminated by neglecting 
the premise ; and for every such premise neglected 

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a term is eliminated. In regard both to related 
and unrelated premises and their terms, the neglect 
of all irrelevant terms and premises may be con- 
sidered a process of elimination which accompanies 
all thought. 

37. Inference is judgment of judgments, and Science of 
ascertains the sameness of samenesses. Science, 

When in comparing A with B, and the same B 
with C, we judge that A=B and B=C, we ob- 
tain sciencej or reasoned knowledge of things, as 
distinguished from the mere knowledge of sense 
or feeling. But when we judge the judgments 
A=:B=C to be the same, as regards A and C^ 
with the judgment A=C, we obtain Science of 

Here is the true province of logic, long called 
Scientia Scientiarum. Hence it is that logic is 
concerned not with the truth of propositions 
per S6 (§ 20), but only with the truth of one as 
depending on others. 

SCIENCB OF SCIBKCB {A«H=C} = {A«C} Beasonino 
SciENCB A=B B«C Judgment 

Things ABC Apprehension 

38. Here we find the clear meaning of the Form and 
distinction of form and matter of thought. matter. 

Sameness of Samenesses » Form ) \ 

Sameness of things - j ^^^^ \ [ Of thought 

Things « Matter)] 

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Addition 39. In discourse, when several names are 

%ia8^^^' placed together side by side, the meaning of the 
joint term is sometimes the sum of the meanings 
of the separate terms.* 

So in our system, when two or more terms are 
placed together^ the joint term must have as its 
meaning the sum of the meanings of the separate 
terms. These must be thought of together and 
in one. 

* I shall here consider only the cases of combina- 
tion in which the combined term means the added meaU' 
ings of the separate terms. The same forms of reasoning 
apply, as I believe, mutatis mutandis^ to any cases of 
combination imder some such wider law as this — 

Same parts samdy related make same wholes. 

Only by some such extension can logic be made to 
embrace the major part of all ordinary reasoning, which 
has never yet been embraced by it, save so far as this 
may have been done in some of Professor De Morgan's 
latest writings. But to show how such an extension may 
be grafted on to my system must be reserved for a future 
opportunity. In most relations it is obvious that the order 
of terms in relation is no longer indifferent. (§ 41.) 

Concerning some inferences by combination, see Thom- 
son's Outlines^ §§ 87, 88. 

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40. Any terms placed together will be said to Combina- 
form with respect to any of those separate terms 1^^ 

a combination or combined term. With respect to 
all other terms they may be called simply a 
term. For it must be remembered that any 
single term, A, B, C, &c., is not more single 
in meaning than a combination. 

41. The meaning of a combination of terms is Order of 
the same in whatever order the terms be combined. J?^ indif- 

Thus, AB = BA; ABCD = BACD = ferent. 
DCAB, and so on. 

For the order of the terms can at most affect 
only the order in which we think of them, and 
in things themselves there is no such order of 
qualities and circumstances. (Boole, p. 30.) 

42. A combination of a term with itself is the Law of 
same m meaning with the term alone, '^ ^ 

Thus AA = A, AAA = A, and so on. 

Also, a combination of terms is not altered by 
combination with the whole or any part of itself. 
Thus ABCD = ABCD . BCD=A . BB . CC . DD 
=ABCD, since BB = B, CC = C, DT> = D. 

The coalescence of same terms in combination 
must be constantly before the reader's mind. 

This important and self-evident law of logic 
was first brought into proper notice by Prof. 
Boole (p. 32), who remarks : * To say " good, 
good," in relation to any subject, though a cum- 
brous and useless pleonasm, is the same as to say 
" good." ' 

Professor Boole gave to this law the name 
Law of Duality. But as this name, on the one 
hand is not peculiarly, adapted to express the 

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Degree of 

Law of 

parts and 

by com- 

general feet AAAAA = A, and is pecu- 
liarly adapted to express the fact A = AB + Ah 
(§ 99), I have yentured to transfer the name, and 
substitute a new one. 

43. In the terms as used under the above law 
there is no reference to degree of quality. When 
required, each degree of quality may be treated 
in a separate term, containing as part of its mean- 
ing every less degree of the quality. Two or 
more degrees of a same quality in logical com- 
bination therefore produce the greatest of those 

44. ■ It is in the nature of thought and things 
that ie;Aen same qualities are joined to same qualities 
the wholes are same. 

Hence the law of logic — 

Same terms combined with same terms give same 
combined terms. 

Thus, since A = A and B = B, therefore 
AB = BA = AB. 

This self-evident law is a more general case of 
Euclid's second axiom. It may, perhaps, be most 
briefly stated as follows i^-Same parts make same 

48. Same terms being combined vnth both mem- 
bers of a premise, the combinations may be stated 
as same in a new proposition which will be true 
with the premise. 

For what is true of terms obviously the same, 
as A, A, or B, B, must also be true of terms 
known to be the same in meaning by a premise. 
Thus, from A =s B we may infer AC = BC 
by combining C with each of A and B. 

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As the number of possible terms which may be 
combined with the terms of a premise is infinite, 
there may be drawn from any premise an infinite 
number of inferences by combination. 

46. Inferences which may be drawn by com- Combiner 
bining the members of two or more premises need ^^.^fP^' 
not be considered here. 

47. A proposition inferred by combination Gmend 
(§ 45) will be true with its premise, whatever be J^^ ^^ 
the term or terms used for combination. When f&rences. 
terms of specific meaning, indeed, are selected at 
random, it will usually happen that the combina- 
tions of the inference are imheard-of, absurd, and 
useless. This does not affect the truth of the in- 
ferred proposition, which only asserts that the 
meaning of the one combination, whatever it be, 

is the same as the meaning of the other. - 

48. In our daily use of specific terms, we con- Tacit rela- 
stantly use each under the restriction of a number ^t^, J^" 
of premises so well known to all persons that it 

is needless to express them. Terms joined not in 
accordance with these tacit relations make non- 
sense. For instance, the impassable difference of 
matter and mind renders it nonsense to join the 
name of any material with that of any mental at- 
tribute, except in a merely metaphorical sense. 
In order, then, that our inferences should always 
be intelligible and useful, we should require the 
expression of all tacit premises connected with 
terms of specific meaning. It is only the several 
branches of science, however, that can undertake 
the necessary investigations in detail. Our infer- 
ence remains true, however complicated be the 

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of common 

tion cU' 

relations of sameness and difierence of the terms 
introduced. But it is inference from premises 
which are stated^ not from those which migM he 
or ought to he stated, • 

49. When premises contain terms only par- 
tially the same, the combination of each with the 
part that is different in the other will produce a 
term completely the same in each. Such premises 
may be considered as related. (§31.) 

Thus, in A = C and B = CD, the terms 
C and CD are only partially the same. But 
the combination of D with A = C gives AD 
= CD, having one member completely the same 
as one member of B = CD. Hence we may 
infer AD = CD = B (§ 26), and eliminate the 
term C, which was common in the premises : 
thus, AD = B. 

Again, to eliminate B from the premises A 
= BC and E = BD, combine D with each 
side of the first, and C with each side of the 
second. Hence, AD = BCD = CE, or AD 
= CE, in which B does not appear. 

60. From premises which have no term in 
common, this process will only give us the in- 
ferences which might be had (§ 46) by the direct 
combination of the respective terms of the pre- 
mises. Thus, A = B and C = D give 
AD = BD, and BC = BD, whence AD = 
BC. And we might similarly get AC = BD. 

61. The following process may be called suh- 
stitution, and will be seen to give the same 
inference as the two processes of forming a 
common term (§§49, 27), and then eliminating it. 

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F&r any temtj or part-term, in one premise, may } » / 
be suhatituted its expression (§ 29) in other terms. ^ * 

In short, the two members of any premise may . , 
be used indifferently, one in place of the other, 
wherever either occurs. 

Thus, if A=BCD and BC=E, we may 
in the former premise substitute for BC its 
expression E, getting A=DE. The fiill process 
of inference consists in combining D with both 
sides of BG=E, and eliminating the complete 
common term BCD thus obtained, so that 

52. We may substitute for any part of one intrinsic 
member of a proposition the whole of the other. elindna- 

Thus, in A=BCD, we may substitute for 
any one of B, C, D, BC, BD and CD, parts of 
BCD the one member, the whole, A, of the other 
member, inferring the new propositions — 


The validity of this process depends on the ) 
Laws of Simplicity (§ 42), and of Part and / 
Whole (§ 44), as is seen by combining each mem- 
ber of the premise with itself. Thus, from 
A=BCD we have A.A=BCD.BCD=BCD.D 
=AD by coalescence of same terms, and sub- 
stitution for BCD of its expression A. 

The new proposition thus inferred will have 
one of its sides pleonastic, that is, with some part ■ 
of its meaning repeated. But it is obvious that 
we cannot, as a general rule, substitute for part of 
one side less than the whole of the other, because 

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we cannot from the premise alone know that the 
meaning of the part-term removed is quite sup- 
plied in the part of the other member put for it. 

The above process may be called intrinsic eli- 
minatiouy to distinguish it from the former process 
of elimination between two premises, which may 
be called extrinsic elimination, and is seen to be 
that case of intrinsic elimination in which we 
substitute for the whole of one side the whole of 
the other. In a single premise, intrinsic elimina- 
tion of a whole member would give only an 
identical and useless result. 

Intrinsic elimination gives no new knowledge, 
but is of constant use in striking out or abstract- 
ing terms concerning which we do not desire 
knowledge, and which are therefore worse than 
useless in our results. 

Professor Boole's system of elimination (p. 99), 
is, I believe, equivalent to the above, though the 
correspondence may not at first sight be apparent. 
Failure of 53. A term cannot be intrinsically eliminated 
^ti^^^' which occurs in both members of a proposition. 
The presence of such part-term may be called a 
condition of the sameness of the remainder of the 

54. Terms are said to be samely related in a 
premise when their interchange does not alter the 

Thus, B and C are samely related in A=BC, 
because the premise is the same A=GB (§ 41) 
after their interchange. But A and B are not 
samely related, because their interchange altera 
the premise into B=:AC. 



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In A=BCDE . . . any two of B, C, D .... 

are samely related and may be interchanged. 

55. Of samely related terms, an expression Inference 
for the one is the same as the expression for an- ^^^^^ 
other after the two terms in question have been 

66. When several terms are samely related, Chncem- 
we obtain the expressions concerning the rest »^*^^^^- 
from the expression for any one by successively 
changing each term into the next when the terms 
are kept in some fixed order. 

It is evident that we may always interchange 
the terms in any part of a problem, provided we 
do so throughout the problem (§ 14). And in 
those cases in which the premises remain un- 
changed thereby, we evidently get several infer- 
ences from the same premises. This method of 
interchanges is familiar to mathematicians. 

57. It will be obvious that a mathematical Mathema- 
term or quantity of several factors is strictly ana- f*^ ^^^" 
logons in its laws to a logical combined term, 
excluding the Law of Simplicity, 

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Law of 58. It is in the nature of thought and things that 

^andmrts '^^^fi'^^^ *^^^ ^^^* of qualities same qualities are 

taken, the remaining sets are the same ; or, more 

briefly — Same parts from same wholes leave same 


Hence the logical law: — When from same com- 
binations of terms same terms are taken, the re- 
maining terms are the same. 

This is the converse of the Law of Same Parts 
and Wholes (§44), and is equally self-evident 
with it. But it is not equally useftd with it ; and 
in Pure Logic, in &ct, is of no use at all. The 
removal of terTtis with their known meanings is 
not equally possible with their combination, and 
in useful logical premises, is not possible at all. 
For, in a useful premise (§ 19), a part at least- of 
one member must be unknown, and this part may 
or may not contain the part we desire to remove. 
Even supposing then that a term occurs on either 
side of a premise, we cannot remove it from the 
known side, because we cannot know whether or 
not we can remove it from the imknown or par- 
tially known side. 

Thus in AB^AC, suppose A and G known, 

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and B iinknown. We cannot infer B=C, because 
B may contain part or the whole of the known 
meaning of A, in addition to the known meaning 
of C, by the Law of Simplicity (§ 42), and in 
leaving B, we do not remove A from one member 
of the premise. 

69. The logic of known and unknown terms, ConvpUu 
it has been said (§ 15), is analogous to the calculus J^^^^y jv 
of known and unknown numbers. mathema- 

So, a logic in which all terms were known ^*^*' 
would have an analogue in common Arithmetic, a 
calculus in which all the numbers employed are 
known. Combination of terms has an analogue 
in multiplication of numbers, and separation 
of terms in division of numbers. As in logic 
combination is tmrestricted, so in calculus is 
multiplication. As in logic of known terms 
only, separation of terms is unrestricted, so in 
a calculus of known numbers only, division is 
unrestricted. But, as in logic of known and 
unknown terms separation is restricted^ so in 
calculus of known and unknown numbers division 
is restricted. 

60. It is well known that, in like manner, we Restriction 
cannot divide both sides of an equation by an ^Z^*^'***^- 
unknown fector, and assert the resulting equation 
to be necessarily true, because the unknown 
fector may be = 0. Thus, from xy = xz, we 
cannot remove x, and assert y =^ z, because if 
X happen to be = 0, the equation xy = xz is 
true, whatever finite numbers be the meanings of 
^and z. 

The correspondence is thus shown : — 

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Parts not 
fr&in the 

of terms 
and pre- 

Logical Fbopostttons. 

TerTTis known admit 



(unless either diyidend contain 


Terms unknown admit 


but do not admit 


Mathematical Equatioits. 

Numbers known admit 



(unless diyisor =s 0) 

Numbers unknown admit 


but do not admit 


The above, analogies did not escape the notice 
of Professor Boole (pp. 36-37), and I am therefore 
at a loss to understand on what ground he asserts 
that there is a breach in the correspondence of the 
laws of logic and mathematics. 

61. From the meaning of a whole term we can- 
not learn the meaning of a part. 

In A=BC, if we know A we learn BC as a 
whole; but we do not thence learn the parts 
B, C, separately. For of the qualities in A any 
part may be in B, and any part in C, including 
any part of those in B, by the Law of Simplicity 
(§ 42). It is only necessary that every quality in 
A shaU be either in B or in C. Even if we know 
one of B and C, we only learn of the other that 
it must contain any quality of A not in the first 

We here meet the imperfection of an inverse 

62. With reference to the relation between the 
number of premises, and the numbers of known 
and unknown terms (§§ 33-35), we must treat as 
separate terms any which occur separate in pre- 
mises, although they may also occur in combina- 
tion. Otherwise, we always treat any whole 
combination as a single term. 

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63. A plural term has one of several meanings^ Terms of 

hut it is not known which, ^n^,\,^o 


Thus B or C is a plural term, or term of 
;iiany meanings, for its meaning is either that of 
B or that of C, but it is not known which. 

A term not in form plural, may be distinguished 
as single ; such is A. 

64. The separate terms expressing the several Alternative 
possible meanings of a plural term are called aZ- 
temativesy and are to be joined together by the 

sign + placed between each two adjoining terms. 

All that has been said of single terms applies to 
plural terms, mutatis mutandis. 

66. The meaning of a plural term is the same Order of 
whatever be the order of the alternatives. ^ws^' 

Either B or G is the same in meaning as 
either C or B, that is, B + C=C + B. For the 
order in which we think of the possible qualities 
of a thing cannot alter those qualities, and the 
order must not convey any intimation that one 
meaning is more probable than another. 

66. A term is combined with a plural term by Combiner 
combining it with each of its alternatives. ^^^ 

For what is A and either B or C, if it is B, term. 

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Use of 

tion of 

Law of 


is AB ; if it is C, is AC, and it is therefore 
either AB or AC. 

67. Let a plural term enclosed in brackets 

( ), and placed beside another term, 

mean that it is combined with it, as one single 
term is with another : 

Thus A (B -f C) = AB + AC. 

68. One plural term is combined with another 
by combining each alternative of the one separately 
with each of the other. Each combined alter- 
native may then be combined with each alternative 
of a third plural term, and so on : 

Thus(DH-E)(B + C)=B(D+E) +C(D + E) 
=BD + BE + CD-|-CE. 

69. It is in the nature of thought and things 
that same alternatives are together same in meaning^ 
as any one taken singly. 

Thus, what is the same as A or A is the same 
as A, a self-evident truth. 
A+A=A A-|-A+A=A A+A+B=A+B 

This law is correlative to the Law of Simplicity, 
(§ 39), and is perhaps of equal importance and 
frequent uee. It was^not recognised by Professor 
Boole, when laying down the principles of his 

70. In a plural term, any alternative may be re- 
moved, of which a part forms another alternative. 
Thus the term either B or BC is the same in 
meaning with B alone, or B + BC=B. For it 
is a self-evident truth (§99) that B standing alone 
is either the same as BC, or as B not-C. Thus 
B+BCs=B not'C+BC+BC 
=B no^C+BC=B. 

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71. A plural term obeys the Law of Simplicity Plural 

For let A=B -h C ; then — single 

AA=(B+C)(B-hC). '^^'^• 

AA=BB-fBC+BC+CC (§68). 

A= B+BC+C (§42). 

A= B + C (§70). 

A plural term obeys the Law of Unity (§ 69) : 
A+A=B + C+Bh-C = B-i-C. 

72. For any alternative or part of an alternative Subsiiitt- 
may be substituted (§ 51) its expression in other ^^^^^ 
terms : tenm. 

Thus, if A=B+CD and D=E, substitute, 
getting A=B + CE. 

78. A plural term may be substituted like a Suhstitu- 
singleterm for any term, single or plural, of which J^^.^ 
it is the expression. When in combination, the terTna. 
several alternatives must be separately combined 
(§§ 66, 68). 

Conversely, for a plural term may be substituted 
its expression in a single term : 

Thus, if A=BC and C=D4-E, for C sub- 
stitute D+E, and A=B (D+E)=BD4-BE. 

Or from the premises A=BD+BE= 
B(D+E) and C=D-hE, we might by substitu- 
tion get back to A=BC. 

74. A plural term is known when each of its Plural 

alternatives is known. ^f^ 


Thus, in A=B -|- C, A is known when the 
meanings of each of B and C are known. But 
of course from knowing a single meaning of A, 
we cannot learn either or both of B and C. 

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Ntmher of 76. With reference to the relation between the 

terms and number of premises, and the nmnbers of known 
premises. _ _ ^ ,«.«.«« *»,-v 

and imknown terms (§§ 33-35), we must treat 

as a separate term each alternative of a plural 

A proposition with a plural term thus corre- 
sponds to an equation with several unknown 
Plural and 76. As plural terms obey the laws of single 
^i^gl^ terms, and a term single in form may be plural in 

meamng, it will not be necessary for the future to 
distinguish plural arid single termSy any more 
than it has been to distinguish combined and 
simple terms. 

There is some danger of misconception con- 
cerning plural terms. Though a plural term has 
one of several meanings, it cannot bear in this 
system more than one at the same time, so to 
speak. Hence it still remains a unit^ the name of 
a single set of qualities, one of several sets, but it 
is not known which. The whole of this system 
in short is unitary ^ and involves the same remark- 
able analogies to a calculus of unity and which 
have been brought forward so explicitly in Professor 
Boole's system. 

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Terms may also be known and stated as differ- 
ing, or not being the same in meaning. 

77. It is in the nature of thought and things Law of 
that a thing which differs frcm another differs ^€^^^^* 
from everything the same as that other. 

More briefly stated — Same as different are 

Hence in logic — 

A term which differs from another term in 
meaning differs from every term which is the same 
as that other. 

If A is not the same as B, which is the same 
as C, then A is not the same as C. The infe- 
ren^ce arises in the sameness of B and C, allowing 
us to substitute one for the other. Hence we learn 
nothing of the sameness or difference of any two 
terms, D and E, each of which differs from a 
third, F ; for D and E may each have any of an 
indefinite variety of meanings, and each may yet 
differ from F. (§152.) 

78. Hence a chain of related premises between Neaative 
any of which inferences can be drawn, must not ^^fi^^<^^' 
contain more than a single negative premise. 

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Law of 
parts and 

Law of 

Also any inference in whidi a negative premise is 
concerned must be a negative inference. 

79. A negative proposition is simply convertible. 
For A is not the same as B, is the same state- 
ment as JB is not the same as A. 

80. When same terms are combined with diffe- 
rent termSy the wholes may be different 

If A differs from B, then AC differs from 
BC, provided, however, that the difference of A 
and B does not consist in any part of C. 

81. When from different wholes same parts are 
taken, the remainders are different. 

This is equally self-evident with the preceding 

It is unnecessary further to consider negative 
propositions, because their inferences may be ob- 
tained by use of aflirmative propositions. 

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82. The known meaning of a negative term is Negative 
the absence of the quality^ or set of qualities^ which ^^^' 
forms the known meaning of a certain other, itspo' 

sitive term. 

Thus not'A is the negative term signifying 
the absence of the quality or set of qualities A, 
If the known meaning of A be only a single 
quality, not-A means its absence ; but if A mean 
several qualities, not-A means the absence of 
any one or more. 

Thus, if ^=5.(7 

not'A = B not'C + not-B.C + not-B not-C. 

83. The negative of a negative term is the cor^ Negative of 
responding positive term. negative. 

What is not-not'A is A. 

84. Since the relation of a positive to a negative Simple 
term is the same as the relation of a negative to a <^^^^'^ 
positive, let each be called the simple contrary fined, 
term of the other. 

85. For convenience let not'A be written a. Notation. 
Then any large and its small letter denote a pair 

of simple contraries; and not-a is A. Also, 
the contrary of BC (§ 82) is 

Be -^ bG + be. 

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' Involve * 

of plural 


Law of 



tory term 

which expresses the absence of one or more of B 

86. All that has been said of a term applies 
samely to one as to the other of a pair of contraries. 

Thus, a obeys the several laws : 

C=:D J 

aa^=^a a-^a^a p___ jv and so on. 

87. Let a combined term or a proposition be 
said to involve a term when it contains either that 
term or its contrary. 

88. The contrary of a plural term is a term 
containing a contrary of each alternative. 

Thus the contrary of A + B + C is ahc. If 
any alternative has more than one contrary, for 
each there will be a contrary alternative. Thus, 
A + BC has the plural contrary aBc +ahC-\- ahc. 

89. Any combined tenp which contains the 
simple contrary of another term may be called 
a contrary, or contrary combination of this, or of 
any combination containing this. 

Thus, any combined term containing A is a 
contrary of any term containing a, and it will 
seldom be necessary to distinguish by name simple 
contraries, such as A and a from contraries, or 
contrary conpibinations in general, which merely 
contain A or a. (See, however, §§ 99, 100.) 

90. It is iu the nature of thought and things 
that a thing cannot both have and not have the 
same quality, 

9L Hence a term which means a collection of 
qualities in which the same quality both is and is 
not, cannot mean the qualities of anything which 
is or ever will be known. 

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Such a term then has tw? meaning, that is to say, 
no possible, useful, or thinkable meaning ; but it 
may be said to mean nothing. Let it be called a 
self-contradictory, or, for sake of brevity, a con^ 
tradictory term, 

92. Let us denote by the term or mark , l^se of 0. 
combined with any term, that this is contradic- 
tory, and thus excluded from thought. Then 
Aa=Aa.O, Bi=B6.0, and so on. For brevity 
we may write Aa=0, Bft=0. Such propositions 
are tacit premises of aU reasoning. 

Any two contrary terms in combination give a 
contradictory term. 

98. Ai^ term being combined with a contra- Combina- 
dictoiy, the whole is contradictory. ^JZt^dL 

For the whole then means a collectibn of tory. 
qualities which does and does not contain some 
same quality, and is therefore by definition a con- 

Thus, if A=B6 =BJ.O 

94. The term 0, meaning excluded from thought, Term 0. 
obeys the laws of terms. 

0.0=0 + 0=0, 
otherwise expressed : — ^What is excluded and ex- 
cluded is excluded — What is excluded or excluded 
is excluded. 

96. Any term not known to he contradictory Condition 

must he taken as not contradictory, ' ofnon-con- 

. , 1 ,. . tradiction. 

Any term known to be contradictory is 

excluded from notice, and any term concerning 

which we are desiring knowledge must therefore 

be assumed not contradictory. 

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tory alter- 

Hon of con- 

tion of al- 

96. In a plural term of which not all the alter- 
natives are contradictory^ the contradictory alter* 
native or alternatives must he excluded from 

If for instance A=0+B, we may infer A=B, 
because A if it be is excluded ; and if it be 
such as we can desire knowledge of, it must be 
the other alternative B. 

97. No contradictory term is to he eliminated in 
direct inference. 

For all we can require to know of a contradic- 
tory term is that it is contradictory, and elimina- 
tion of a contradictory term would prevent rather 
than give such knowledge. 

Thus if A=Cc . 0, B=Cc . 0, all that we can 
require to know of A and B is known from 
these premises, and cannot be known from the 
inference A=B got by eliminating the contradic- 
tory Cc. 0. 

So, if A=B=C=D=E=F=G^.O, the only 
useful inferences are those showing each of A, B, C, 
D, E, F, to be contradictory. 

So, also, obviously, of intrinsic elimination. 

It may be said, in fact, that contradiction super- 
sedes all other elimination by itself eliminating all 
contradictory terms from fttrther notice. 

98. An alternative is eliminated when its plural 
term is combined with a contrary of that alterna- 

Thus, the alternative Ab is removed from the 
plural term AB + Ab when combined with B. 

(AB-H Ab)B = AB H- ABb = AB + = AB 

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Let C=AB+Aft+aB + aJ 
BC=AB4-aB AJbC ^Ah 
aC=aB +ah aBC =aB 

5C=:Aft +a6 aftC =aft. 

The term thus combmed with each side cannot 
be eliminated intrinsically (§ 53), and remains a 
condition of the rejection of the other alternative. 

It is by this rejection of alternatives that the 
extent or -width of the meaning of a term is 
reduced, as its intent of known meaning is in- 
creased, by combination (§^ 1). For every general 
term, in addition to its known meaning, may be 
assimied to have an indefinite multitude of unknown 
alternatives. In combination with a new terra 
many of these will probably become contradic- 

D 2 

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Law of 


99. It is in the nature of thought and things 
that a thing is either the same or not the same 
as another thing. Otherwise — 

A set of qualities either does or does not contain ' 
a certain quality. 

Hence, in logic, a term must contain the meaning 
of one of any pair of simple contrary terms. Thus : — 

A term is not altered in meaning by combination 
with any simple contrary terms as alternatives. 

A=A(B + *) = AB+A5. 

For if A has meanings containing only B, then 
A 5 is contradictory, and A=AB + 0=AB. 

If A has meanings containing only b, then 
AB=0 and A=0 + A6=Aft. 

If A has meanings of which some contain B 
and some b, the law is still true. 

This Law of Duality is not the same as Pro- 
fessor Boole's law of duality. (See § 42.) 

100. Some apparent exceptions may occur to 
this law. For instance, let A= virtue, B=:black, 
and 5= not-black. Then the statement 

Virtue is either black or not-bl<zck, seems true 
according to the above law, and yet absurd. This 

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arises from B and h not being simple contraries ; 
for B may be decomposed into black-coloured — 
say BC, and b into not-black-coloured, or not black 
and not coloured, or bC + be. Now, virtue is really 
not coloured at all, or is Abe, and, therefore, 
neither BC nor bC Here, again, we must observe 
that the combination Be is contradictory from the 
tacit premise black is a colour (§ 48). 

Other apparent inconsistencies may be similarly 

Professor De Morgan has excellently said,* * It 
is not for human reason to say what are the simple 
attributes into which an attribute may be decom- 
posed.^ And for such a reason it is that I have 
as &r as possible abstained from treating any 
terra as known to be simple, 

101. Let a term, combined with simple contra- Devdope- 
ries as alternatives, be called a developement of the ^**^ 
term as regards the contraries. 

Thus, AB-j-Ab is called a developement of A 
as regards B, or in terms of B, or involving B. 

102. Any term is same in meaning ajier combi- Continued 
nation with all the possible combinations of other ^^^^^ ^' 
terms, and their contraries as alternatives. 

Since A=AB-i-A5, and, again, A=AC+Ac, 
we may substitute for A in AB+Aft (§ 51) its 
expression in terms of C. Thus, 

A= ABC -h ABc + AbC -h Abe. 

Again, since A=AD+Ac?, we may substitute 
a second time, getting 
A=ABCD4-ABCcZ+ +AJc(^, and so on. 

* 8yllahu8, p. 60. 

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Dual term 103« Let any two alternatives, differing only by 
defined. ^ single part-term and its contrary, be called a 
diuzl term. 

Thus, AB + A5 is a dual term as regards B, 

and ABC-f ABc as regards C, and we may speak 

of B + J or C + c as the dual part. 

Seduction 104. A dual term may always he reduced to a 

%rm^ sm^Ze term by removal of the contrary terms^ mth- 

out altering the meaning. 

For the term thus obtained is, by the Law of 
Duality, the same in meaning as the former dual 
term (§ 99). 

Thus, from such a term as AB+Aft, we may 
always remove the dual part B + ft, and the mean- 
ing of the term A will still be as before, since 
A= AB + Ai is a self-evident (§99) truth always 
in our knowledge. 

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105. From any affirmative premise we may infer Affirmative 
a negative proposition by changing any term on one VJ^ ^^' 
side only into its contrary. position. 

From A=B we have Anot=J; for evidently 
B is not=;J, and hence, by Law of Difference 
(§77), A=Bnot=ft, or A not = ft. 

From AB=AC, similarly, AB not=Ac. 

106. The two terms of a negative proposition Terms of 
are contraries. ^^! 

For the two terms of a negative proposition are tion. 
different in meaning. Hence there must be some 
quality or qualities in the meaning of one, and not 
in that of the other ; thus, the combination of the 
two terms would mean both the absence and pre- 
sence of a certain quality or qualities, and would 
be a contradictory. The two terms then are con- 
trary (§ 89). 

107. A negative proposition may he changed into Negative 
an affirmative, of which one term is d term of the ^!i-fg " 
negative^ and the other term this term combined proposi- 
tcith the contrary of the other term of the negative. ^^* 

Thus, if A not=B, then A=Aft; or, again, 

For developing A in terms of B (§ 101), we 

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have A=AB-hAft, but A and B being contra- 
ries (§ 106), AB is contradictory or 0. Hence, 

Similarly, we may show B = + «B = aB . So, 
if ABnot=AC, then AB=ABc. For AB = 
ABC+ABc=ABc, since ABC is contradictory. 
And we see that 

ABc=AB (contrary of AC) 

=AB (Ac-t-aC-|-ac)=ABc + 0-hO. 

Since we may now convert any negative propo- 
sition into an affirmative, it will not be further 
necessary to use negative propositions in the pro- 
cess of inference. (§81.) 
Inference 108. From any contradictory combination we 

iivTw'opo- '"^y *V^^ ^^^^ ^''^y P^^^ ^f *^^ cotnbination not 

sitions. itself contradictory is not the same in meaning as 
the remainder or any greater part. That the two 
parts differ may be expressed in a negative propo- 
sition, or its corresponding affirmative. 

For if the other part be contradictory, it cannot 
be the same as the first part, which is not contra- 
dictory. And if neither of the parts is contra- 
dictory in itself, they cannot be same in meaning, 
else their combination would not produce a con- 

The affirmative inferences corresponding (§ 107) 
to the negative ones deduced under this rule may 
be otherwise had, so that it seems unnecessary to 
consider the negative inferences further in this 

List of 109. The following are the chief laws or con- 

^^^*- ditions of logic : — 

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Condition or postulate. The meaning of a term 
must be same throughout any piece of reasoning ; 
so that A=A, B=B and so on. (§ 14.) 

Law of Sameness, (§ 25.) 

A=B=C; hence A=C. 

Law of Simplicity. (§ 42.) 

AA=A, BBB=B, and so on! 

Law of Same Parts and WTioles, (§ 44.) 
A=B; hence AC=BC. 

I^w of Unity. (§ 69.) 

A+A=A, B + Bh-B=B, and so on. 

Law of Contradiction. (§ 90.) 

Aa=0, AB5=0, and so on. 

Law of Duality. (§ 99.) 

A=A(B + 6)=AB4AJ 
A=A(B4-*) (C+c) 

=ABC-f ABc-hA6C-|-Aic and so on. 

It seems likely that these are the primary and 
sufficient laws of thought, and others only corol- 
laries of them. Logic may treat only of known 
samenesses of things ; and differences of things need 
be noticed, only for the exclusion from pure logical 
thought of all that is self-contradictory. 

In pure number and its science, on the other 
hand, differences of things only are noticed. 

The Laws of Simplicity, Unity, Contradiction 
and Duality furnish the universal premises of 
reasoning. The Law of Sameness is of altogether 
a higher order, involving Inference, or the Judg- 
ment of Judgments. 

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Tlseofde- 110. Taken by itself, the developement of a 
vdopement ^^^^ (^ ^qj^ ^^^ ^ ^^ ^^^^ knowledge about it. 

But taken with the premises of a problem, we may 
learn that some of the alternatives of the develope- 
ment are contradictory and to be rejected. The 
remaining alternatives then form a new and often 
useiul expression for the term. 
Indirect HI. In thus using a developement we are said 

inference. ^^ ^^^ indirectly y because we use the premise to 
show what a term is, not directly by the Law of 
Sameness, but indirectly by showing what it is not. 
Indirect Inference is direct inference with the 
aid of self-evident premises derived from the Laws 
of Contradiction and Duality. But all Inference 
is stiU by the Law of Sameness. 

112. Let A=B : required expressions for A, B, 
a, b, inferred from this premise. Develope these 
terms as follows (§ 101) : — 

A=AB-hA5 B=AB-f-aB 
rt=aB -\-ab 5=Aft +ab 

Examine which of the alternatives AB, A6, 
aB, ab, are contradictory according to the premise 

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A combined with A=B gives A=AB 

B „ „ „ „ AB=B 

a „ „ „ „ Aa=aB=0 

b „ „ „ „ Aft=B6=0 

Hence we learn tHat dB and Ab are contradic- 
tory, and may be rejected, and that AB is not 
contradictory according to the premise. Of ab^ 
which is not foimd among any of the above terms, 
we can learn nothing from the premise, and it 
therefore cannot be known to be contradictory. 
Striking out dB and Ab in the developements of 
A, B, a, ft, we have — 

A=AB+0 =AB B=AB-hO =AB 
a = -f-aft= ab 5=0 +ab=ab 


113. We have here the two inferences A=AB Inferences. 
B=AB which might have been had from the 
premise by combination (§ 45), and from which 

we may pass back by elimination of AB to the 

We also have a=aft, and ft=saft, which could 
not have been had by direct inference. And by eli- 
minating ab between these two we have the new 
inference a = ft. This result, indeed, that from the 
eameness of meaning of two terins, we may infer 
the sameness of meaning of their simple contranea 
is evidently true. 

114. By a similar method we may draw infer- Inference 
ences from any nxunber of premises, namely, by I^J^J^^ 
developing any required term in respect of other 

terms, and striking out the combinations which 
are shown to be contradictory in any premise. 

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Method of 







Thus, from A=B and B=C, to infer expres- 
sions for A and a, we develope these terms as 

follows : — 


a=aBC+ aBc+ abC-habc 
By combination we then, when possible, render 
one side of each premise same with each of the 
alternative combinations, and learn from the other 
side whether the combination is known to be con- 
tradictory by the premise. All the combinations 
in the above developements will be found contra- 
dictory, except ABC and abcj and we thus get the 
inferences A=ABC, and a=a6c, of which the 
former indeed might have been got directly. 

115. The process of indirect inference may 
similarly be applied to drawing any possible infer- 
ence or expression from any series of premises, 
however numerous and complicated. The full 
process may be abbreviated according to the fol- 
lowing series of rules, which may be said to form 


1. Any premises being given, form a combi- 
nation containing every term involved therein 
(§87). Change successively each simple term of 
this into its contrary, so as to form all the possible 
combinations of the simple terms and their con- 

2. Combine successively each such combina- 
tion with both members of a premise. When the 
combination forms a contradiction with neither 
aide of a premise, call it an included subject of the 
premise ; when it forms a contradiction with both 
sides, call it an excluded subject of the premise ; 


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when it forms a contradiction with one side only, 

call it a contradictory combination or subject^ and C&ntradic- 

etrike it out. ^*^- 

We may call either an included or excluded 
subject a possible subject, as distinguished from a Possible. 
contradictory combination or impossible svhject. Impossible. 

3. Perform the same process with each premise. Eepeated 
Then a combination is an included subject of a '^'^P^^' 
series of premises, when it is an included subject 

of any one ; it is a contradictory subject when it 
is a contradictory of any one ; it is an excluded 
subject when it is an excluded subject of every 

4. The expression for any term involved in the Selection. 
premises consists of all the included and excluded 
subjects containing the term, treated as alterna- 

5. Such expression may be simplified by re- Reduction, 
ducing all dual terms (§ 104), and by intrinsic 
elimination (§ 52) of all terms not required in the 

6. When it is observed that the expression Mimina-^ 
of a term contains a combination which would not ^*^^* 
occur in the expression of any contrary of that 

term, we may eliminate the part of the combina- 
tion common to the term and its expression. (See 
below, § 117,) 

7. Unless each term of the premises and the Contradic- 

contrary of each appear in one or otlier of the ^^^ . 

., / , . ; . . , 1 I premises. 

possible subjects, the premises must be deemed 

inconsistent or contradictory. Hence there must 

always remain at least two possible subjects. 

(§ 159.) 

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tion ex- 

116. Required by the above process the infer- 
ences of the premise A=BC. 

The possible combinations of the terms A,B,C, 

and their contraries, are as given in the margin. 

Each of these being combined with both sides of 

the premise, we have the following results : — 

ABC =ABC ABC included subject 

ABc =ABCc =0 ABc contradiction 

AhQ =AB6C =0 AbC contradiction 

A^c =AB6Cc=0 
0=AaBc =aBCc =0 
0=AaiC=aBiC =0 
0— Aa6c =aB^Cc =0 







excluded subject 
excluded subject 
excluded subject 

It appears, then, that the four combinations 
ABc to aBC are to be struck out, and only the 
rest retained as possible subjects. 

Suppose we now require an expression for the 
term b as inferred from the premise A=BC. 
Select from the included and excluded subjects 
such as contain b, namely abC and abc. 

Then b=iabC-{-abc, but as aC occurs only with 
6, and not with B, its contraiy, we may, by 
Rule 6, eliminate b from abC ; hence ft=aC+a5c. 

117. The validity of this last elimination is seen 
by drawing the expression for aC, which is abC. 
Then between bssiabC-\-abc, and abC=aG, we 
may eliminate abC by substituting (§51) its ex- 
pression aC. And similarly in all other cases to 
which the rule applies. 

We might also reduce the expression for b by 
Rule 5, as follows : — 

b=ahC + abc=iah (C-\-c) = ab. 

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118. To express a we have Other in- 

a=aBc + abC + abc, f"^"^^' 

but observing that none of Be, bC, be, occur with 
A, so that Bc=aBc, 6C=aftC, ic=aftc, we sub- 
stitute these simpler terms, eliminating a ; whence 
asszBc+bC+bCj an evident truth (§ 113). 

119. Similarly, we may draw any of the follow- Other in- 
ing inferences:— ^'^''''''' 





a6=a^C-t-a5c=a5 (no inference) 
ac^aBc-j-abc^ac (no inference). 

120. Observe that since B and C are samely Relation of 
related to A, we may get any inference concerning ■" ^^^ ^' 
one of these terms from the similar inference con- 
cerning the other by interchanging B and C, 

ft and c (§ 56). 

Before proceeding to further examples of in- 
direct inference, we may make the following 

121. When any term appears on both sides of a Excluded 
premise, as A in AB = AC, any combination con- *^^*^^*- 
taining its contrary, a, is an excluded subject. 

Thus, in combining any term with both sides of a 
proposition, we render any contrary of the term 
an excluded subject. 

So, in mathematics we introduce a new root 
into an equation when we multiply both sides by 
a ^tor. 

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122. An excluded subject, though admitting of 
inference and admitted into inferences, is of infe- 
rior and often of no importance. As its name 
expresses, it is usually a combination concerning 
which we do not desire knowledge. The sphere 
of an argument, or the Universe of Thought^ con- 
tains all the included subjects. An excluded sub- 
ject is such as lies beyond this sphere or umverse. 
But we are obliged to consider excluded subjects, 
because the excluded subject of one premise may 
be the included subjfect of other premises. 

123. When a premise is plural in one or both 
sides, an excluded subject is a contrary of all the 
alternatives on both sides, and a contradictory 
combination is a contrary of all on one side, and 
not of all on the other side. 

124. Of an identical proposition the term itself 
appearing on either side is the only included sub- 
ject. All others are excluded, and there are no 
contradictory combinations. Its useless nature is 
thus evident. 

125. Any subject of a proposition remains an 
included, excluded, or contradictory subject as 
before, after combination with any imrelated terms. 
Thus, if the argument be restricted to a sphere or 
common subject^ defined by certain terms, these do 
not need expression in each premise, but may be 
retained as an exterior condition. Thus, by 

ABCD ( ) we might mean that ABCD 

is to be understood as combined with each term 
of any premises placed within the brackets. ABCD 
is then the common subject of the premises, which 
must contain no contrary of this. And any con- 

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trary of ABCD is an excluded subject of tlie 

126. Any set of terms which always occur in Ofun- 
the premises in unbroken combination may be combina- 
treated as a simple term. tions. 

Thus, if BC occur always thus in combination, 
we may write for it, say D, and then d or 
not-BChbC + Bc-^hc. 

127. Any set of alternatives which always occur Unbroken 
together in the premises as alternatives may be P^^^^ 
treated as a single term. 

Thus, if B and C occur always as alternatives, 
we may for B -|- C write, say D, and then d or 
neither B nor C is be, 

128. Any proposition may be treated under the Simple 
form of A=B, so long as we do not require to ?J^^^' 
treat its part- terms or alternatives separately. (By 

129. Hence the convenience in every branch of Technical 
knowledge of using technical terms to stand for 

every large set of terms which usually occur to- 
gether. But such terms become the source of error 
if we do not carefully keep before us their defini- 
tions, those adopted premises in which we express 
the set of combined or alternative terms for which 
we substitute a technical term. 

130. In that branch of knowledge, however, Meta- 
called First Philosojjhy, which is analytic^ and ^J!^ 
aims at resolving things, or our thoughts about 
them, into their simplest components, the use of 
technical terms is ^llacious. Such terms cannot 
assist analysis, since each arises from the synthesis 

of many simpler terms, fonmiTTy its definition. 


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ed process. 


All reasoning, then, in Metaphysics or First Phi- 
losophy, ought to be carried on in the simplest 
and most vernacular elements of speech. Ana- 
lytic science should be like a mill which grinds 
down the ordinary grains of thought into their 
smallest and simplest particles. It is in the bake- 
house we should combine these particles again into 
loaves of a size and consistency suitable for ordi- 
nary use. But most metaphysical reasoners, it 
seems to me, have mistaken the mill and the 

131. It is not always necessary to carry out the 
process of inference exactly as in the rules. Each 
or any premise may be treated as a separate one, 
if desirable, and its possible subjects afterwards 
combined with the possible subjects of other pre- 
mises. We may thus successively add premises, 
or try the effect of supposed ones. 

For instance, since AB and ah are the possible 
subjects of A=B, and BC and he of B=C, the 
possible combinations of these, namely ABC and 
dbc^ are the possible subjects of the two premises 
combined, observing that AB6c and aJBC are 

132. If premises be related, the indirect infe- 
rences will include aU possible direct inferences. 
From imrelated premises we shall also get such 
inferences as are possible. 

Thus, from the unrelated premises 

\ have 

A=B, C=D, 

A=BCD + Bc(Z 

ei=ABc + aJc, and so on. 

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133. It does not seem possible to give any Proof of 
general proof that the conclusions of the indirect *^^T^^ 
method must agree with those of the direct me- 
thod, which will make its truth any the more 
evident Such proof could be little less than 

a general recapitulation of the several Laws of 

134. It hardly needs to be pointed out that the EuclicPs 
method of indirect inference is equivalent to ^^^^^^^^ 
Euclid's indirect demonstration, or reductio ad tion. 
ahmrdum. Euclid assumes the developement of 
alternatives, usually that of equal or greater or 

'less, and showing that two of these lead to a con- 
tradiction, establishes the truth of the third. 

136. Nor is this process of reasoning at all new Common 

or uncommon in any branch of knowledge save ^f«<?/'*w- 
1*1.1 1,1 . ^ ^«^f^^ 

logic, which was supposed to be the science of method. 

all reasoning. Simple instances occur perhaps as 
frequently as instances of direct inference, and 
complicated instances are only rendered scarce by 
the limited powers of human memory and atten- 
tion. Among instances of indirect argument we 
may place all those discourses in which a writer 
or speaker states several possible alternatives or 
cases of his subject, and, after showing some of 
them to be impossible, concludes the rest to be 
necessary, or else proceeds further to develope 
and consider these with regard to other premises 
(§ 131). A good instance is found in Paley's 
Argument on the Divine Benevolence (Moral 
Phil., Book II. chap. V.). The old logical process 
called abscissio infiniti has a close relation to 
indirect inference. 


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Quotation, 136. Even brute animals, it would seem, may 
reason by the indirect method :— 

* This creature, saith Chrysippus (of the dog), 
is not void of Logick : for, when in following any 
beast he cometh to three several ways, he smelleth 
• to the one, and then to the second ; and if he find 
that the beast which he pursueth be not fled one 
of these two ways, he presently, without smelling 
any further to it, taketh the third way ; which, 
saith the same Philosopher, is as if he reasoned 
thus : the Beast must be gone either this, or this, 
or the other way ; but neither this nor this; ErgOj. 
the third : so away he runneth.' 

Sir W. Raleigh, 

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Before giving examples of the processes of 
logical inference as now set forth, it will be well 
to- consider the relation of our system to the logic 
of common thought. 

137. In ordinary reasoning it will be found that Ordinary 
there is great economy of thought. Not only are ^^^^^ 
large collections of attributes and things grouped 
together under the fewest possible terms, but only 

those particular attributes of the things under 
consideration on which the reasoning turns are 
brought forward. A certain natural disinclination 
to exertion causes us to simplify our modes of 
thought as much as possible, and to leave in the 
background everything that is not essential. Thus 
when we say man is mortal^ we mean that the at- 
tributes of mortality are among the attributes of 
man. But we leave out those infinitely nume- 
rous attributes of man which are not comprised 
under mortality, because we do not happen to be 
occupied with them. The proposition, then, in 
this form is not that equation of qualities, that 
statement of perfect sameness or equivalence of 
meaning, which we have taken as a proposition. 

138. It may be objected that we ought to take 

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or same- " 
neas the 
true form 
of reason- 

tion of pre- 

the proposition as we find it in common thought. 
Aristotle so took it, and his system has had a long 
reign. Some of the expounders of his system 
even denied that there could be a proposition of 
two universal and equivalent terms. They could 
not have committed a greater error or more com- 
pletely misrepresented the ordinary course of 
reasoning. Not only, as a fact, do the several 
sciences establish multitudes of propositions of 
which the two terms are equivalent and universal, 
but all definitions are propositions of this kind, 
and the definitions requisite in connecting the 
meanings of more and less complex terms, must 
always form a large part of our data in reasoning. 
If we iurther consider that even Aristotle's nega- 
tive propositions have a universal predicate, that 
men show a constant tendency to treat the predi- 
cate of the proposition A as universal, whence 
several common kinds of fallacy, and that reasoning 
from same to same things may be detected as 
the fundamental principle of all the sciences, 
we need have no hesitation in treating the equation 
as the true proposition, and Aristotie's form as 
an imperfect proposition. 

It is thus the Law of Sameness, not the dictum 
of Aristotie, which governs reasoning. 

139, It is only of veiy late years that the im- 
perfection of the ordinary proposition has been 
properly pointed out. It is the discovery of the 
so-called quantification of the predicate which has 
reduced the proposition to the form of a con- 
vertible equation, and opened out to logic an 
indefinite field of improvement. 

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140. Professor Boole's system, first published Bool^s 
in his ^Mathematical Analysis of LogiCy in '-^^^Sf^* 
1847, involves this newly discovered quantifica- 
tion of the predicate. According to Boole, the 

some J which is the adjective of particular logical 
quality, is an indefinite class symhoL Men are 
some mortals is expressed by him in the equation, 
x=vy, where x instructs us to select from the 
universe all things that are men, and y to select 
all things that are mortal. The proposition then 
informs us that the things which are men consist 
of an indefinite selection from among the things 
which are mortal, v being the symbol of this in- 
definite quantity or class selected. 

141. One more step seems to me necessary. It Further 

is to separate completely the qualitative and ^^ '^^^" 
quantitative meanings of all logical terms, in* 
eluding the word some. In the qualitative form 
of the proposition man is some mortal — or more 
correctly speaking, man is some kind of mortal — ^we 
interpret some or some kind as meaning an in- ' 
definite and perhaps unknown collection of quali- 
ties, which being added to the quahties mortalj 
give the known qualities of num. In the quanti- 
tative form men are some mortals^ we have the 
equivalent statement that the collection of indi- 
viduals in the class som^ mortals is the collection 
of individuals in the class men, 

142. It is strange that the purely qualitative Qtuditaiive 
form of proposition man is some kind of mortal j F^^^P*^^ 
which is the most distinct form of statement, and ded, 

is perhaps the most prevalent, both in science 
and ordinary thought^ was totally disregarded by 

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logicians, at least as the foundation of a system 
of logic. ' The Logicians, until our day,' says 
Professor De Morgan,* * have considered the ex- 
tent of a term as the only object of logic, under 
the name of the logical whole ;' the intent was 
called by them the metaphysical whole, and was 
excluded from logic' 
'Some' 143. It will be seen that this word some or 

^kind/^^^ 50W6 kind, the source of so much difficulty 
and error, must in our system be treated as a 
term of indefinite and imknown meaning. It is 
an unknown term, not only at the beginning of a 
problem, but throughout it. In no two premises 
then can the term some or some kind be taken to 
mean the same set of qualities. Thus we cannot 
argue through or eliminate a term with some, 
while at least it retains this unknown term : that 
is to say, we can never use it as a common term 
(§ 27) in direct inference. Thus, if A is some B, 
and some B is some C, we cannot eliminate some B 
' getting A is some C, because some being of 
imknown meaning, the some B is not necessarily 
the same in both cases. This is still more plain 
in the form A is some kind of B, and some kind 
ofB is some kind of C, for it is obvious that the 
one kind of B is not necessarily the same as the 

144. Since the term some or some kind is not 
only unknown but remains unknown throughout 
any argmnent, we might conveniently appropriate 
to it some symbol such as U, to remind us of its 
Special conditions. Thus no term U is to be taken 
!^, p. 61. 

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as same with any other term U, or U=U is 
not known to be true. But in the propositions A 
and E it is always open to us, and is best to eli- 
minate U by writing for it the other member of 
the proposition (§ 52). Thus, A=UB, meaning 
that A is some kind of B, involves three terms. It 
is much better written as A=AB, involving 
only A and B, and yet perfectly expressing that 
the qualities of B are among those of A, but not 
necessarily those of A all among those of B. 

146. The four propositions of the old logic may Aristotle's 
thus find expression in our system : — proposi- 

A=UB or A=AB 

A=Uft or A=Aft 


UA=Uft or CA=D6. 

146. Two new propositions of De Morgan's DeMor- 

system are thus expressed : — gansjpro- 

'' ^ posittons. 

Everything is either A or B A=b 

Some things are neither A nor B a =ib. 

147. All these propositions, and as many more Thomson^s 
as may be proposed, can be brought and partially P^'oposi- 
treated (§ 128) under the form A=B, which I 
believe to be the simple form of all reasoning. 

The existence of doubly-imiversal propositions of 
this kind was far from being unknown to many 
of the School Logicians, but out of deference 
to the Aristotelian system, such propositions were 
neglected. The present Archbishop of York first 
embodied this proposition in a system of logic, 
giving it the name U. (Thomson's * Outlines,' 

A . 

Every A is B 

E . 

, No AisB 

I . 

Some A is B 


Some A is not B 

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In this chapter I shall place some miscellaneous 

examples of inference according to the system of 

the foregoing chapters, suited to show the power 

of its method, or its relation to the old logic. 

. . 148. Let us take a syllogism in Felapton. 

in Fdap- '' ° 

ton. No A is B A=U5 = A5 

Every A is C A=UC=AC ^ 

SomeC is not B (§ 145.) 

Direct From A=Aft we might by combination (§ 45) 

infer&me, infer AC = AftC, and from A = AC, A6=AftC; 

whence AC = A6C^A5, or AC = A5, which is 

a more precise statement of UC = TJb, or some 

C is not B, the Aristotelian conclusion. 

We may, however, obtain this conclusion, as 
well as all other possible ones, by indirect infer- 
AbC Of the possible combinations of A, B, C, a, b, c, 

aBC ABC and ABc are contradicted by the first pre- 
oBc mise, and Abe (as well as ABc) is contradicted by 
«*C the second premise. A6C, aBC, aBc, abO, ahc, are 
o-bc the remaining combinations in which we find there 
is no relation between B and Cper se, since B occurs 
with C and c, and C occurs with B and b» But 

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AC = A5C, and A& = A&C, whence, by elimina- 
tion, AC = A6, the same conclusion as before. 
The following conclusions may also be drawn : 

a = a (BC -h Be + 5C + 5c) = a (B -h h) (C + c) 

= a (no inference) 
B =aB 

6 r= AC H- ahO + ahc = AC + a6 

C = A5 + aBC -H abO = A6 + aC 

c = aBc + ahc = ac 
a5 = ahC H- aJc = aft (C + c) = ah (no inference) 
aC=aBC + ahC = aC (B + 6) =aC (no inference) 
he = ahc, 

149. The premise AB = CD is of some interest. 
It contradicts the combinations ABCc?, ABcD, 
ARcd^ which are AB and not CD, and A6CD, 
aBCD, aftCD, which are CD and not AB. From 
the remainder we easily draw the inferences 

A= BCD -h A5C(Z + A5cD + Aftcc? 
a=BCc;+BcD +Bc(^ +a6C(^-f aftcD-f-aftce?. 

Observing that A and B enter samely into the 
premise, we may easily deduce the expressions 
for B and h by interchanging A and B, a and h 
in the above ; thus (§§ 54-56) — 

B = ACD + dBCd -h aBcD + d&cd. 

And since A, B enter samely with C, D, we 
might deduce the corresponding expressions for 
C, D, and c, a, by interchanging at once A with 
C, and B with D, or A with D, and B with C. 

From the expression for a we thus get 
d = CBa + CftA + Qha + dcRa + dchA + dcha. 

Observe, that if the expression for A be com- 

Example : 


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bined with that for a, nothing but contradictory 
terms will be the result, verifying Aa=0. And, 
if we combine the expressions for any terms not 
contrary, as B and d, we get the same result 
as we might have drawn by the separate applica* 
tion of the process. 

Thus, B(Z= AJCD + dBQd + a^cd= + oBd. 

In expressions thus derived there will often 
appear, as in the above instance, a superfluous 
and contradictory term (A5CD, a contrary of Be?), 
but being only an alternative, the proposition is 
not untrue. 
Example : 150. As an example of a premise with a plural 
^^^""^ term, let us take A=B-fC. 

In comparing the eight combinations of A, B, C, 
ABC a^ Jj c, with the premise, any one is contradictory 
^^ which contains A without containing either B or C ; 
or, again, which contains either B or C without 
containing A. Thus, ABC, ABc, and A6C, are 
the included subjects, dbc is an excluded subject, 
and the rest are contradictory. 
We may draw the inferences 



6=rA6C +a 
C=ABC+A5 = AC 

c=ABc +a. 

Observe that B and C enter samely, so that 
their expressions may be mutually derived by 



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161. The premise As=Bc+5C differs from the Example: 
last in the very important point that A cannot T^ 
at once be B and C. 

It has the included subjects ABc and AbCy ABc 
and the excluded subjects aBC and abc. The A60 
following expressions are seen to be simple and ^^^ 
symmetrical, and it is instructive to form their ^ 

A=Bc +hG 


B=Ac H-aC 


C=AJ +aB 

c=AB + aJ. 

162. From two negative premises we can infer Negative 
no AristoteHan conclusion (§§ 77, 78). It is weU P^^^^^^ 
to show that this remains true when the negative 

, propositions are converted into their corresponding 
affirmatives (§ 107). 
Let us take the premises 

A is not the same as B, 
C is not the same as B. 

These may be expressed by the affirmative 
propositions As=A5, C=6C. 

If we go through the process of indirect in- AbC 

ference, and attempt to express A and C in ^-g^ 

terms of each other, we shall obtain : — abO 

A^AbC+Abc=:Ab (C+c)=Aft 

C=A6C+a5C=iC (A-ho)=iC. 

These are the premises over again, and there can 
be no new inference, except B=aBc. ^ 

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of form 

















of two pre- 


163. The proposition A=B being the simplest 
form of statement, its foil solution is given below, 
and the solutions of the similar propositions A= J, 
a=B, a=5, are inferred by interchanging A and a, 
B and b. 

Premise A = B 

Included subject AB 
Excluded subject ab 
Contradiction Ab 

Contradiction aB 

154. Let us take A=ABC, 


We have, by direct inference from the second 
premise, BC=BCD (§ 45) 

Hence A=ABC=ABCD (§ 26.) 

The indirect process gives four included and 
two excluded subjects, as in the margin. 

Hence not only the above inference, but the 
following, among other possible ones :— 

a=aBCD+aBcD +abCD-\'abcD-\-abcd 

=aBD +a5D +abd 

=aD -i-abd =aBD +ab 
c=aBcD -f «JcD ^abcd =acD -^abd 
Bc = aBcD cD=acD 

165. The ordinary Sorites is easily and clearly 
solved in this system. Taking four premises such 
as A=AB 



D=DE, many inferences will be evident 

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from the following series of the subjects, or pos- 
sible combinations. 







^Included subjects. 

Excluded subjects. 

166. The Dilemma of the old logic is easily Dilemma. 
included in our system, when we supply a term 
which is suppressed or imderstood in its usual 
statement. The dilenmia is as follows : — 

If A is B, E is F, and if C is D, E is F: 
but, either A is B, or C is D, therefore E is F. 
Adopting Wallis's reduction to the categorical 
form, and supplying some term G, to express 
the present circumstances, or the case in which 
either A is B, or C is D, we have the pre- 


By the direct process alone we get the required 
conclusion that, under the condition G, E is F ; 
thus — 

or, GE=GEF. 

157. The following is known as a Destructive Destructive 
Ck)nditional Syllogism. Tm^ 

If A is B, C is D ; but C is not D ; there- 
fore, A is not B. 

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Forms of 
old logic. 


Supplying the suppressed term, say E, express-* 
ing the circumstances in which A is not B, the 
following is the statement of this syllogism in our 
system : — 

By direct inference 

Hence ABE is known to be contradictory ; there- 
fore (§ 108), AE is not ABE, or in the circum- 
stances E, A is not B. 

168. The forms of the old logic being compre- 
hended in this system along with an indefinite 
multitude of other forms, logicians can only pro- 
perly accept this generalisation, due to Boole, by 
throwing off as dead encumbrances the useless dis- 
tinctions of the Aristotelian system. The past 
history of the Science must not, as hitherto, bar 
its progress. And Logic will be developed almost 
like Mathematics, when Logicians like Mathema- 
ticians discriminate between the Study of Thought 
and the Study of Antiquarian Lore. 

I will now give a few complex problems, more 
suited to show the power of the method. 
159. Let the premises be 
B=c -^d 
AD = BCD. 
And let it be required to infer the description 
of any term, say a. By the indirect process, we 
shall find that the only combination uncontra- 
dicted by one or other premise is ABCc?. 

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Thus, we find there cannot be any a at all, 
without contradiction, whatever may be the mean- 
ing of this result.* It means, doubtless, that the 
premises are contradictory. (§ 115.7.) 

* The following law, being of a lees evident character Law of in- 
than the rest^ has been placed apart finity. 

Every logical term miist have its contrary. 

That is to say: — Whatever quality we treat as present 
we may also treat as absent. 

There is thus no boundaiy to the universe of logic. Universe of 
No term can be proposed wide enough to cover its whole ^J^"' ?^' 
sphere ; for the contrary of any term must add a sphere ^^ 
of indefinite magnitude. Let U be the universe ; then u\b 
not included in U. Nor will special terms limit the universe. 

Thing existing has its contrary in thing not existing. 

Thing thinkable has its contrary in thing not thinkable. 

Even thing, the widest noun in the language, has a 
contrary in that which is not a thing. 

Of course the above is only true speaking in the strictest 
logical sense, and using all terms in the most perfect 

If the above be granted as true, every proposition of the Contradic • 
form A =B + 6 must be regarded as contradictory of a law torypropo- 
of thought. For the contrary of A firom the above is **^*^^* 
B6, a contradiction, or A is used as having no contrary, 
and forming the universe. 

Also every system of premises must be rejected which Contradic- 
altogether contradicts any term or terms. Thus in the ^^ V^^' 
indirect process we must always have at least two combina- ^**^* 
tions remaining possible, one of which must contain the 
contrary of each simple term in the other. In this view 
the peculiar premises 

A«B + C 
B«c +d 
(§ 159) contain subtle contradictions. For a according 
to the first premise must be he, and being c, it must 
by the second premise be B, and hence by first premise 
also, A, or both A and a, B and b. 

But this subject needs more consideration. 

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We also easily infer any of the following : — 








160. The following premises are such as might 
easily occur in physical science : — 

A=ABc +AbC 
B=BDE +Bde 

The series of possible combinations in the mar- 
gin gives by inspection perhaps the most useM 
information, but the following are a few formal 

A= ABcDE + ABc^e + AhCBe 
cd= ABcde +aBcde +abcd(EA-e) 
bCD -AbCDe -{•abCDe 

There is no relation between abc and D and E. 
161. I conclude with the solution of a still 
more complicated system of premises. 

A4-C + E = B+D + F 

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The possible cqmbinations are : — 

ABcDe/ aBceZEF 

ABcc^EF o^cdEf 

AJ^cdEf aJCc?EF 

Whence the following, among many other in- 
ferences, may be drawn : — 

A=BcDe/-f ABccZE + AftCcffiF 

rfE=ABcc^ + JCF+aBc=Bc6?-h6CF 
AF= ABc^F 4- AJCe^EF 
aF=aBc(ZEF +a^>C(ZEF 
h =«>C^EF 

Whence the remarkable and unexpected rela- 
tion C=6, which it would not be easy to detect 
in the premises. 

162. Inferences may be verified by combining Verifica- 
the expressions of two or more terms, and com- ^*^* 
paring the result with the expression of the 
combined term as drawn from the series of pos- 
sible combinations. For instance, in the problem 
last given (§ 161), we may c«mbine the expres- 
sion for A with that for 6?E, as follows : — 
A . dE=(BcDe/+ ABcc^E 4- A^>Cc^EF)(Bc^-h ^>GF) 
=04- ABc^E + + + A. JCc^EF, 

the contradictory combinations being struck out. 

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But the expressions thus obtamed may not always 
be in their simplest terms. 
Sedttction 163. The reduction of inferences to their 
Mry?*^^" simplest terms, it may be remarked, is in no way 
essential to their truth; it only renders them 
more pregnant with information. It is, perhaps, 
the only part of the process in which there is any 
Worhingof 164. In working these logical problems, it has 
the process, i)een found very convenient to have a series of 
combinations of terms beginning with those of 
A, B, and proceeding up to those of A, B, C, D, 
E, F, or more, engraved upon a common writing 
slate. In any given problem, the series is chosen 
which just furnishes sufficient letters for the dis- 
tinct terms. The contradictory combinations may 
then be rapidly struck out, and the remaining 
combinations lie ready before the eye. 

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166. To show the power and facility of this 
method, as compared with that of Professor Boole, 
it will be sufficient, as regards those abready ac- 
quainted with Professor Boole's system, to present 
the solution of one of his complex examples. Thus, 
let us follow Professor Boole's investigation of 8emof& 
Senior's definition of wealth, namely* — that wealth ^^^^ 
18 what is transferable, limited in supply , and either 
productive of pleasure or preventive of pain, 
(Boole, p. 106.) 

Let A = Wealth 

B = Transferable 
C=Limited in supply 
D=Productive of pleasure 
E=Preventive of pain. 

The definition in question is expressed by the 


which includes all the combinations of D, E, 
d, e, except de. 

* Here, as usually elsewhere, I take words in intent of 
meaning, and transform most statements accordingly. 

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Striking out the dual term (E-f e) from BCD 
(E-f e), we may state the definition in the more 
concise form 

A=BCD + BCdE. 

We may pass over Prof. Boole*s expression for 
A, after intrinsic elimination of E (A=BCD 
-f ABCef), as being sufficiently obvious. 

166. . Required C in terms of A, B, D (Boole, 
p. 107). 

Forming all the possible combinations of A, B, 
C, D, E, and their contraries, and comparing them 
with the premise, we shall find all the combina- 
tions firom ABCde to aBCdE inclusive contra- 
dicted. The remaining subjects are as in the 

Selecting the terms containing C, we have 
C= ABCDE + ABCDe + ABCc?E + aBCde 
+ a5CDE +abCDe -\-ahCdE -{-abCde. 

Striking out the dual terms (E -|- e), and intrin- 
sically eliminating remaining E's or e's by substi- 
tution of C, we have 

C=ABCT>+ABCd+aBCd+abCD + abCd. 

Eliminating C fix)m ABCD (§ 117), because 
ABD=ABCD, and striking out the dual terms 
(A+a) and (B+d), we have either of the ex- 
pressions — 

C=ABD + BCc? +abC 

From the latter we read, WTiat is limited in 
supply is either wealthy transferable (and either 
productive of pleasure or not, ABC), or else some 

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.kind of what is not wealthy hut is either not trans- 
ferable (abC), or, if transferable, is not productive 
of pleasure {oBCd), 

This conduaioii is exactly equivalent to that of 
Professor Boole, on p. 108. 

167. His so-called secondary propositions, Negative 
namely, * 1. Wealth that is intransferable and pro- ^^^^^ 
ductive of pleasure, does not exist ; ' and * 2. 
Wealth that is intransferable and not productive 

of pleasure does not exist,' are negative conclu- 
sions implied in the striking out of the contra- 
dictory combinations A5CDE, A5CD«, AftcDE, 
AftcDe, and AftCe^E, AJbQde, AbcdE, Abcde, 
which are easily reducible to 

A5D (C-f c) (E+e)=0 A6D=0 
Abd (C + c) (E + e)=0 Abd =0. 

The expression * does not exist ' is open to ex- 

168. Again, required an expression for produc- 
tive of pleasure (D), in terms of wealth (A), and Expremm 
preventive of pain (E). (Boole, p. 111.) 

The complete collection of combinations con- 

taining D is 









We may then write D as follows : — 
But we may observe also that 


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Hence we may substitute ADE and Ae for the 
two first terms of the expression for D. We may 
also strike out the dual term (E+c) in the third 
term, and eliminate the plural term (Bc + ftC + ftc) 
intrinsically (§ 52) by substitution of D. Thus 
we get the expression in the required terms : 


which may be translated into these words : — Whc^ 

is productive of happiness is either some kind of 

wealth preventive of pain, or any hind of wealth 

not preventive of pain, or some hind of what is 

not wealth, (Boole, p. Ill ad fin,) 

Expression 169. For the expression of d we easily select 
for d. 

<?=Ac?E+arfe+ac?E=ArfE + «d 

of which the meaning is — What is not prod\ictive 
of pleasure is either some hind of wealth preventive 
of pain, or some hind of what is not wealth. 
(Boole, p. 112.) 
Other in' 170. These are the chief inferences furnished 
ferences. ^^ |^ Boole. From the list of possible combi- 
nations we could easily add a great many more 
inferences, in iact as many more, as may be drawn 
concerning any of the five terms A, B, C, D, E, 
and their contraries. 

Thus for CE expressed in the remaining terms, 
we have 

==(ABCE + abCE) (D + d). 

Striking out the dual term (D-f-rf) and extrin- 
sicaUy eliminating C in ABCE, since we observe 
that ABE=ABCE, we have 

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which may be translated — 

What is limited in supply, and preventive of 
pain, is either wealth, transferable and preventive of 
pain, or some hind of what is not wealth and not 

But ive may oflen find that there is no special 
relation to express. Thus, in trying to express 
aftCD in terms of E we find 

aftCD=aftCD (E + c)=a6CD. 

171. Besides affording these formal .deductions, General 
the series of possible combinations will often give f^^^^ 
us at a glance a clear and valuable notion of the nations. 
manner in which the universe of our subject is 

made up. 

In this instance we see that for wealth we have 
the three combinations BCDE, BCDe and 
BGdE, and that thus for not-wealth (a) we have 
all possible combinations of B, C, D, E, except 
those three. With aB we have Cde and 
€ (DE + D6-f-^E+<?e), and with ab, we have all 
possible combinations of C, D and E. Thus the 
definition gives no relation between what is not 
wealth and not transferable, and what is limited 
in supply, productive of pleasure, or preventive of 

172. It is the character of this logical system, Generality 
in common with that of Professor Boole, that it is ^f ^^ ^*" 
perfectly general. The same rules which govern 

the inferences fi'om one or two premises, involving 
two or three terms, are applicable without the 
slightest modification to any number of premises. 

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involving any ntunber of terms. Of course the 

working of the inferences becomes rapidly more 

laborioiis as the complexity of the problem increases, 

and a considerable liability to mistake arises. 

But this is in the nature of things, and the process 

of inference, consisting in the mere comparison of 

terms as to their sameness or difference, seems to 

me the simplest process that can be conceived. 

Convpari- 173. Compared with Professor Boole's system, 

BooI^s ^^ ^^ mathematical dress, this system shows the 

si/stem. following advantages. 

1. Every process is of self-evident nature and 
force, and governed by laws as simple and primary 
as those of Euclid's axioms. 

2. The process is infallible, and gives no imin- 
terpretable or anomalous results. 

3. The inferences may be drawn with far less 
labour than in Professor Boole's system, which 
generally requires a separate computation and de* 
velopement for each inference. 

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1 74. So long as Professor Boole's system of mathematical 
logic was capable of giving results beyond the power of 
any other system, it had in this feet an impregnable strong- 
hold. Those who were not prepared to draw the same 
inferences in some other manner could not quarrel with 
the manner of Professor Boole. But if it be true that the 
system of the foregoing chapters is of equal power with 
Professor Boole's system, the case is altered. There are now 
two systems of notation, giving the same formal results, 
one of which gives them with self-evident force and mean- 
ing, the other by dark and symbolic processes. The burden 
of proof is shifted, and it must be for the author or sup- 
porters of the dark system to show that it is in some way 
superior to the evident system. 

175. It is not to be denied that Boole's system is con- 
sistent and perfect within itself. It is, perhaps, one of the 
most marvellous and admirable pieces of reasoning ever 
put together. Indeed, if Professor Ferrier, in his Institutes of 
Metaphysics, is right in holding that the chief excellence of 
a system is in being reasoned and consistent within itself, 
then Professor Boole's is nearly or quite the most perfect 
system ever struck out by a single writer. 

176. But a system perfect within itself may not be a 
perfect representation of the natural system of human 
thought. The laws and conditions of thought as laid down 
in the system may not correspond to the laws and condi- 
tions of thought in reality. If so, the system will not be 

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one of Pure and Natural Logic. Such is, I believe, the 
case. Professor Boole's system is Pure Logic fettered with a 
condition which converts it from a purely logical into a 
numerical system. His inferences are not logical inferences ; 
hence they require to be interpreted, or translated back 
into logical inferences, which might have been had without 
ever quitting the self-evident processes of pure logic. 

Among various objections which I might urge to Boole's 
system, regarded as purely logical in purpose, are four 
chief ones to which I shall here confine my attention. 

First Objection, 

177. Boole's symbols are essentially different from the 
names or symbols of common discourse — his logic is not the 
logic of common thought. 

Professor Boole uses the symbol + to join terms together, 
on the understanding that they are logical contraries, which 
cannot be predicated of the same thing or combined 
together witiiout contradiction. He says (p. 32) — * In 
strictness, the words " and," " or," interposed between the 
terms descriptive of two or more classes of objects, imply 
that those classes are quite distinct, so that no member of 
one is found in another.' 

178. This I altogether dispute. In the ordinary use 
of these conjunctions, we do not necessarily join logical 
contraries only ; and when terms so joined do prove to be 
logically contrary, it is by virtue of a tacit premise, some- 
thing in the meaning of the names and our knowledge of 
them, which teaches us they are contrary. And when our 
knowledge of the meanings of tfee words joined is defec- 
tive, it will often be impossible to decide whether terms 
joined by conjunctions are contrary or not. 

179. Take, for instance, the proposition — * A peer is 
either a duke, or a marquis, or an earl, or a viscount, or a 
baron.' If expressed in Professor Boole's symbols, it would be 
implied that a peer cannot be at once a duke and marquis, 
or marquis and earl. Yet many peers do possess two or 
more titles, and the Prince of Wales is Duke of Comyrall, 
Earl of Chester, Baron Renfrew, &c. K it were enacted 

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by parliament that no peer should have more than one 
title, this would be the tacit premise which Professor Boole 
assumes to exist. 

Again, — * Academic graduates are either bachelors, mas- 
ters, or doctors ' does not imply that a graduate can be only 
one of these; the higher degree does not annul the 

Shakespeare's lines, — 

* Beauty, truth, and rarity, 
Grace in all simplicity, 
Here inclosed in cinders Ke. 

# * # » 

To this urn let those repair 
That are either true or feir,' — 

certainly do not imply that beauty, truth, rarity, grace, and 
the true and fair are incompatible notions, so that no instance 
of one is an instance of another. 

In the sentence — ' Eepentance is not a single act, but a 
habit or virtue,' it cannot be implied that a virtue is not a 
habit ; by Aristotle's definition it is. 

Milton has the expression in one of his sonnets — 

* Unstain'd by gold or fee,* where it is obvious that if the 
fee is not always gold, the gold is a fee or bribe. 

Tennyson has the expression * wreath or anadem.' Most 
readers would be quite uncertain whether a wreath may be 
an anadem, or an anadem a wreath, or whether they are 
quite distinct or quite the same. 

From Darwin's * Origin,' I take the expression, * When 
we see any part or organ developed in a remarkable degree 
or manner.' In this, or is used twice, and neither time 
disjunctively. For ii part and organ are not synonjmoua, 
at any rate an organ is a part. And it is obvious that a 
part may be developed at the same time both in an extra- 
ordinary degree and manner, although such cases may be 
comparatively rare. 

180. From a careful examination of ordinary writings, 
it will be foxmd .that the meanings of terms joined by * and ' 

* or' vary from absolute identity up to absolute contrariety. 
There is no logical condition of contrariety at all, and 
when we do choose contrary expresflions, it is because our 

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subject demands it. The matter, not the form of an ex- 
pression, points out whether terms are exclusive. And if 
there is one point on which logicians are agreed, it is that 
logic is formal, and pays no regard to anything not formally 
expressed. (See § 48.) 

181. And if a further proof were wanted that Professor 
Boole's symbols do not correspond to those of language, we 
have only to turn to his own work. He actually translates 
one same sentence into different sets of symbols, according 
to the view he takes of the matter in hand. For instance, 
(p. 59) he interprets * Either productive of pleasure or 
preventive of pain ' so as not to exclude things both pro- 
ductive of pleasure and preventive of pain. * It is plain,' 
he remarks, ' from the nature of the subject, that the ex- 
pression " either productive of pleasure or preventive of 
pain," in the above definition, is meant to be equivalent to 
" either productive of pleasure ; or if not productive of 
pleasure, preventive of pain." ' 

And in remarking upon other possible interpretations, 
he says, * that before attempting to translate our data into 
the rigorous language of symbols, it is above all things 
necessary to ascertain the intended import of the words we 
are using.' (p. 60). This simply amounts to consulting the 
matter, and Professor Boole's symbols thus constantly imply 
restrictions not expressed in the forms of language, but 
existing, if at all, as tacit or understood premises. 

182. In my system, on the contrary, I take A+B not 
to imply at all that A may not be B, but if this be the 
case, it must be owing to an expressed premise A= AJ or 

183. How essential Professor Boole's restriction on his 
symbols is to the stability of his system, one instance 
will show. Take his proposition on p. 35 — 

and give the following meanings to a:, y, z : — 

a; = Caesar 

3/= Conqueror of the Gauls 

z= First emperor of Rome. 

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Now, there is nothing logically absurd in saying * Gsesar is 
the conqueror of the Grauls or the first emperor of Rome/ 

It is quite conceivable that a person should remember 
just enough of history to make this statement and nothing 
more. And there is nothing in the logical character of 
the terms to decide whether the conqueror could or could 
not be the same person as the first emperor. 

But now take Professor Boole's inference from the propo- 
sition aj=y -fz, namely x — -?=y got by subtracting z from 
either side of x^=y + z. Then we have the strange infer- 
ence: — 

CcBsary provided he is not the first emperor of Eomej is 
the conqueror of the Gauls, 

This leads me to my second objection to Professor 
Boole's system. 

Second Objection, 

184. There are no such operations as addition and sub^ 
traction in pure logic. 

The operations of logic are the combination and separa- 
tion of terms, or their meanings, corresponding to multi- 
plication and division in mathematics. I cannot support 
this statement without going at once to the gist of the 
whole matter. 

185. Number, then, and the science of number, arise 
out of logic, and the conditions of number are defined by 
logic. It has been thought that units are units inasmuch 
as they are perfectly similar. For instance, three apples 
are three units, inasmuch as each has exactly the same 
qualities as the other in being an apple. The truth is 
exactly opposite to this. Units are units inasmuch as they 
are logically contrary. In so far as three apples are exactly 
like each other, one could not be distinguished from the 
other. Were there three apples, or any three things, so 
perfectly similar in every way that we could not teU the 
difference, they would be but one thing, just as, by the law 
of imity before stated, A+A+A=A. But then we must 
remember that among the logical characters of a thing is 
its position in space with relation to other things, not to 
spe£^ of its position in time. Now, when we speak of 

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three apples, we mean three things, which, however per- 
fectly same they may be in all other qualities, occupy 
different places, and are therefore distinct things. In so 
far as they are same they are one ; in that they are dif* 
ferent they are three, 

186. The meaning of an abstract unit is something only 
known as logically distinct from or contrary to other things. 
The meaning of a concrete unit is the abstract unit with 
certain qualities known or defined. 

For instance, in A (1'4-1" + 1"0=A'+A" + A'" the 
meaning of the units 1', 1", 1"\ is that each is something 
logically distinct from the other , and when we predicate 
of each of these that it is A, say an apple, we get three dis- 
tinct A's, A' + A'' + A'". 

So in multiplication, ttoice two is four — 
(1 + 1)(1 + 1)=1 + 1 + 1 + 1. 

The logical significance of the process is that if we have 
two logically distinct notions, and we divide each into two 
logically distinct notions, we get four logically distinct 
notions. In logical formulsB (A + o) (B + ^)= AB + Aft -f 
aB + ah, where A and a, B and 6, express logical contraries. 

187. Now addition, subtraction, multiplication, and di- 
vision, are alike true as modes of reasoning in numbers, 
where we have the logical condition of a unit as a constant 
restriction. But addition and subtraction do not exist, and 
do not give true results, in a system of piure logic, free 
from the condition of number. 

For instance, take the logical proposition — 


Meaning what is either A or B or C is either A or D 
or Ej and vice versa. 

There being no exterior restrictions of meaning whatever, 
except that the same term must always have the same 
meaning (§ 14), we do not know which of A, D, K, is B, 
nor which is C; nor, conversely, do we know which of A, 
B, C, is D, nor which is £. The proposition alone gives us 
no such information. 

In these circumstances, the action of subtraction does not 
apply. It is not necessarily true that, if from same (equal) 

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things we take same (equal) things the remainders are same 
(eqiial). It is not allowable for us to subtract the same 
thing (A) fix)m both sides of the above proposition, and 
thence infer — 


This is not true if, for instance, each of B and C is the 
same as £, and D is the same as A, which has been taken 

Yet the equivalent inference by combination will be 
valid. We may combine a with both sides of the propo- 
sition, and we have 


or, striking out the contradictory terms oA, we have 

aB+aC=oD + «E. 

188. But subtraction is valid under the logical restriction 
that the several alternatives of a term sh^ be mutually 
exclusive or contrary. Let 

(1) AMN+BMn+CwN=AMN+DMn+EmN 

in which it is obviously impossible that AMN can be either 
DMti or Et/iN, contraries of AMN, or any one of the three 
alternatives any other. Then we may freely subtract 
AMN from both sides, getting the necessaiy inference 

(2) BMw+CwN=DM»+EwN. 

This subtraction, however, is merely equivalent to the 
combination with both sides of the proposition (1) of the 
term (Mw+wN); for the combination being performed, 
and contradictory terms struck out, it will be found that 
the proposition (2) results. 

189. In short, when alternatives are contraries of each 
other, subtraction of one is. exactly equivalent to combina- 
tion with the rest. The axiom (Boole, p. 36), that * if equal 
things are taken from equal things, the remainders are 
equal,' is nothing but a case of the Law of Combination 
(§ 44), that if same (equal) terms be combined with same 
(equal) terms, the wholes are same (equal). 


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Take the self-eyident proposition 

AB+Aft+aB + a&=AB-fAft+aB+a5 

Any terms, say aB + aft, may be subtracted from both 
sides by combining the other terms AB+Aft with each 
side of the proposition. Then 

(AB+AJ) (AB + Aft+aB + a^) 

=(AB+ A^)(AB + Aft + aB+aft) 
AB+AJ+0 . . +0=AB+Aft+0-|- . . 0+0. 

And what is true of this self-evident case must be true 
when the premise is not self-evident. 

190. Having thus established our liberty to subtract 
same terms, provided all alternatives are contraries, we have 
the corresponding liberty to add by the inverse process. 

191. The processes of addition and subtraction thus arise 
out of the logical process of combination. The axioms of 
addition and subtraction are only valid under a logical con- 
dition, which is certainly not applicable to thought or 
language generally. And this condition is that which logic 
imposes upon number, that each two imits shall be contrary 
logical alternatives. It is logic which reduces to a imit, 
by the Law of Unity, A+A=A, any two alternatives 
known to be the same, so that the science of number treating 
of units, treats of alternatives known to be different or con- 
trary. But logic itself is the superior science^ and may treat 
of alternatives of which it is not known whether they are 
same or different, 

192. It is the self-evident logical Law of Unity, then, which 
lays the foundations of number. This law merely amounts 
to sajring that a thing cannot and must not be distinguished 
from itself. We commit an error against this law, when in 
counting over coins, for instance, to ascertain their numbers, 
that is, how many logically distinct coins there are, we 
count the self-same coin two or more times, making the 
coins for instance 


instead of C' + C" + C'" + C''"-|- It is by the 

Law of Unity that C" + C"=C", or the same coin counted 

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twice is but one coin in number. In this case no attention 
is paid to differences of time ; but in many cases, things 
otherwise perfectly the same, like the beats of a pendulum, 
are distinguished and made into different units by one 
being before or after the other in time;. 

Third Objection. 

193. My third objection to Professor Boole's system is, 
that it is inconsistent with the self-evident law of thoughty 
the Law of Unity, (A+A=A.) 

Prof. Boole having assumed as a condition of his system 
that each two terms must be logically distinct, is imable to 
recognise the Law of Unity. It is contradictory of the 
basis of his system. The term a;, in his system, means all 
things with the qvMity x, denoting the things in extent, 
while connoting die quality in intent. If by 1 we denote 
all things of every quality, and then subtract, as in num- 
bers, all those things which have the quality a?, the re- 
mainder must consist of all things of the quality not-x. 
Thus, a?+(l — x) means in his system all x's with all not-x^s, 
which, taken together, must make up all things, or 1. But 
let us now attempt by multiplication with x, to select 
all x's from this expression for all things, 


Professor Boole would here cross out one +x against one 
— a?, leaving one -fa?, the required expression for all x^s. 
It is surely self-evident, however, that x+x is equivalent 
to X alone, whether we regard it in extent of meaning, as 
all the x's added to all the x's, which is simply all the x^s, 
or in intent of meaning, as either x or x, which is surely x. 
Thus, x+x—x is really 0, and not a:, the required 
result, and it is apparent that the process of subtraction in 
logic is inconsistent with the self-evident Law of Unity. 

194. It is probable, indeed, that Professor Boole would 
altogether reflise to recognise such an expression as 
x+x—x, on the ground that it does not obey the condition 
of his symbols that each two alternatives shall be distinct 
and contrary, x+x not being so. It may be answered, 


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that the expression has been arrived at by operations 
enounced as universally valid, which ought to give true 
results. And if it be simply said that x-\'X'^x is not 
interpretdble in Professor Boole*s system, it may be again 
answered, that when .translated into its equivalent in words, 
the expression x-^-x—x has a very plain meaning. It is 
* either x qr x, provided it be not a?,' and this, I must hold, . 
is simply not x, although it ought to be Xy according to the 
mode in which it was got. 

195. In founding his system, Boole assumed that there 
cannot be two terms A + B, the same in meaning or names 
of the same thing ; the laws of thought require nothing of 
the kind, and cannot require it, because among known and 
unknown terms, any two such as A+B may prove to be 
names of the same thing AB. Thought merely reduces 
the meaning of two same terms AB -j- AB, by the Law of 
Unity, to be the same as that of one term AB. And when 
it is once known that all terms in question are contraries of 
each other, or naturally exclusive and distinct, then Boole's 
system and the whole science of numbers apply. 

196. It is on this account that my objections have no 
bearing against Professor Boole's system as applied to the 
Calculus of Probabilities, so far as I can understand the 
subject. For it is a high advantage to that calculus to have 
to treat only events mutually exclusive, probabilities being 
then capable of simple addition and subtraction.* It 
seems likely, indeed, that this distinction of exclusive 'and 
unexclusive alternatives is the Gordian knot in which all 
the abstract logical sciences meet and are entangled. 

Fourth Objection, 

197. The last objection that I shall at present urge 
against Professor Boole's system is, that the symbols -J^, §,^, ^, 
establish for themselves no logical meaning, and only bear a 
meaning derived from some method of reasoning not con-' 
tained in the symbolic system. The meanings, in short, are 
those reached in the self-evident indirect method of the 
present work. 

* See De Morgan's Syllabus, § 243. 

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198. Professor Boole expreesljr allows, as regards one of 
these symbols at least, ^, that it is not his method which 
gives any meaning to the symbol. It is the peculiarity of 
his system, that he bestows a meaning on his symbols by 
interpretation. The interpretation of ^ is explained on 
pp. 89, 90, and he says, ' Although the above determina- 
Jiion of the significance of the symbol -§■ is founded upon 
the examination of a particular case, yet the principle in- 
volved in the demonstration is general, and diere are no 
circumstances under which the symbol can present itself to 
which the same mode of analysis is inapplicable.' Again 
(p. 91), ' Its actual interpretation, however, as an ind.efinite 
dass symbol, cannot, I conceive, except upon the ground 
of ansiogy, be deduced from its aridimetical properties, 
but must be established experimentally.' 

199. If I understand this aright, it simply means, that 
wherever a term appears in a conclusion with the' symbol 
^ afiixed, we may, by a mode of analysis, by some process 
of pure reasoning apart from the symbolic process from 
which ^ emerged, ascertain that the meaning of ^ is 
some, an indefinite class term. The symbol ^ is unknown 
until we give it a meaning. Before, therefore, we can 
know what meaning to give, and be sure that this meaning 
is right, it seems to me we must have another distinct and 
intuitive system by which to get that meaning. Professor 
Boole's system, then, as regards the symbol ^, is not the 
system bestowing certain knowledge ; it is, at most, a sys- 
tem pointing out truths which, by another intuitive system 
of reasoning, we may know to be certainly true. 

200. It is sufficient to show this with regard to a single 
symbol %, because the incapacity of a system, even in a 
single instance, proves the necessity for another system to 
support it. I believe that the other symbols, \y ^, ^, are 
open to exactly the same remarks, but from the way in 
which Mr. Boole treats them, involving the whole condi- 
tions of his system, it would be a lengthy matter to explaii^. 

201. The obscure symbols -J-, ii h \i ^*v® *^® follow- 
ing correspondence with the forms of the present system. 
\ appearing as the coefficient of a term means that the 
term is an included subject of the premise, so that, if 

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combined with both members of the premise, it produces 
a self-contradictory term With neither side (§ 115). 

Similarly, % means that the term is an excluded subject 
of the premise j producing a self- contradictory with both 
sides of the premise. . 

And either ^ or f means that the term is a aelf-covtra- 
dietary or impossible term, producing a self-contradictory, 
term with one side only of the premise. 

208. The correspondence of these obscure forms with the 
self-evident inferences of the present system is so close and 
obvious, as to suggest irresistibly that Professor Boole's 
operations with his abstract calculus of 1 and 0, are a mere 
counterpart of self-evident operations with the intelligible 
symbols of pure logic. Professor Boole starts from logical 
notions, and self-evident laws of thought; he suddenly 
transmutes his formulae into obscure mathematical counter- 
parts, and after various intricate manoeuvres, arrives at 
certain forms, corresponding to forms arrived at directly and 
intuitively by ordinary or Pure Logic — ^by that analysis, 
from which the interpretation of his symbols was reached 
and proved. And by this interpretation he transfers the 
meaning and force of pure logical conclusions to obscure 
forms, which, if they have meaning, have certainly no de- 
monstrative force of themselves. Boole's system is like the 
shadow, the ghost, the reflected image of logic, seen among 
the derivatives of logic. 

803. Supposing it prove true that Professor Boole's Cal- 
ciQuB of 1 and has no real logical force and meaning, it 
cannot be denied that there is still something highly 
remarkable^ something highly mysterious in the &ct, that 
logical forms can be turned into numeral forms, and while 
treated as numbers, still possess formal logical truth. It 
proves that diere is a certain identity of logical and numerical 
reasoning. Logic and mathematics are certainly not inde- 
pendent. And the clue to their connection seems to consist 
in distinct logical terms forming the units of mathematics. 

804< Things as they appear to us in the reality of nature, 
are clothed in inexhaustible attributes, set as it were in a 
frame of time and space. By our mental powers we ab- 
stract first time, then space, and then attribute ailer attri- 

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bute, until we can finally think of things as abstract units 
deprived of all attributes, and only retaining the original 
logical condition of things, that each is distinct from others. 
In logic we argue upon things as same and one, in number 
we reason upon them as distinct and many. 

206. Supposing it be ultimately allowed that Professor 
Boole's calculus of 1 and is not really logic at all ; that his 
system is foimded upon one condition, that of exclusive 
terms, which does not belong to thought in general, but 
only numerical thought; and that it ignored one law of 
logic, the Law of Unity, which really distinguishes a logical 
from a numerical system — ^these errors scarcely detract 
from the beauty and originality of the views he laid open. 
Logic, after his work, is to logic before his work, as mathe- 
matics with equations of any degree are to mathematics 
with equations of one or two degrees. He generalised 
logic so that it became possible to obtain any true inference 
from premises of any degree of complexity, and the work 
I have attempted has been little more than to translate his 
forms into processes of self-evident meaning and force. 

Owens College, ]\1a.nchestei% : 
November, 1863. 




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